A  RECENT  ACHIEVEMENT  OF  APPLIED  PHYSICS 
Andre  Beaumont  speeding  over  the  Marconi  wireless  station  at  Genoa 


A  RECENT  ACHIEVEMENT  OF  PURE  PHYSICS 

Instantaneous  photographs  by  C.  T.R.  Wilson.  1.  Tracks  of  Alpha  par- 
ticles of  radium  through  air.  2.  Tracks  of  Beta  particles  through  air. 
3.  Tracks  of  electrons  ejected  by  X  rays  from  air  molecules  (see  p.  424) 


A  FIRST  COURSE  IN 
PHYSICS 


BY 


ROBERT  ANDREWS  MILLIKAN,  PH.D..  Sc.D. 

H 

PROFESSOR  OF   PHYSICS  IN  THE   UNIVERSITY  OF  CHICAGO 

AND 

HENRY  GORDON  GALE,  PH.D. 

ASSOCIATE   PROFESSOR  OF  PHYSICS  IN   THE   UNIVERSITY  OF  CHICAGO 


REVISED  EDITION 


GINN  AND  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 


COPYRIGHT,  1906,  1913,  BY 
ROBERT  A.  M1LLIKAN  AND  HENRY  G.  GALE 


ALL  RIGHTS   RESERVED 
614.7 


GINN   AND  COMPANY  •  PRO- 
PKJliTOKS  •  BOSTON  •  U.S.A. 


PREFACE 

The  chief  aim  of  the  first  edition  of  this  book  was  to  present 
elementary  physics  in  such  a  way  as  to  stimulate  the  pupil  to  do 
some  thinking  on  his  own  account  about  the  hows  and  whys  of 
the  physical  world  in  which  he  lives.  With  this  end  in  view,  we 
abandoned  the  formal,  didactic  method  so  largely  in  use  during 
the  preceding  decade.  In  place  of  it  we  used  a  method  which 
uniformly  started  with  some  simple  experiment  or  some  well- 
known  phenomenon.  The  consideration  of  how  a  thing  hap- 
pened was  then  followed  by  a  discussion  of  why  it  happened, 
definitions  being  in  general  inserted  only  after  the  need  for 
them  had  been  felt  by  the  pupil.  Finally,  a  carefully  chosen 
set  of  questions  and  problems  following  each  day's  work,  rather 
than  each  chapter,  led  the  "student  to  find  for  himself  the  connec- 
tions between  the  phenomenon  in  hand  and  other  familiar  hap- 
penings. Such  a  method  led  inevitably  to  the  final  grouping 
of  the  apparently  disconnected  facts  of  physics  about  certain 
great  underlying  principles,  such  as  the  kinetic  theory,  the 
work  principle,  the  electron  theory,  and  the  wave  theory. 

Since  the  appearance  of  the  book  we  have  been  deeply 
gratified  both  by  the  fact  that  our  method  has  met  with  con- 
tinually increasing  favor  because  of  its  demonstrated  peda- 
gogical efficiency,  and  by  the  fact  that  the  developments  of 
the  past  seven  years  in  the  science  of  physics  itself  have  so 
amply  justified  our  points  of  view.  For  example,  within  that 
time  the  fundamental  conceptions  of  both  the  kinetic,  theory 
of  matter  and  the  atomic,  or  electronic,  theory  of  electricity 
have  become  as  well  established  as  the  theory  of  the  rotatioil 
of  the  earth  upon  its  axis,  so  that  to  rfail  to  utilize  them  now 
in  the  interpretation  of  the  phenomena  of  molecular  physics 

iii 


iv  PREFACE 

and  of  electricity  would  be  precisely  like  teaching  sunrise  and 
sunset  as  merely  observable  facts  without  any  reference  to 
their  interpretation  in  the  light  of  the  earth's  rotation. 

In  the  present  revision  our  former  method  has  been  main- 
tained. In  addition,  we  have  aimed  to  bring  the  subject  matter 
thoroughly  up  to  date,  and  to  make  certain  changes  which 
seemed  desirable.  The  most  important  of  these  changes  are 
as  follows: 

(1)  The  approach  to  the  subject  of  physics  has  been  made 
more  simple  and  more  interesting  by  postponing  the  chapter 
on  force  and  motion  until  after  the  discussion  of  the  fasci- 
nating phenomena  of  liquids  and  gases. 

(2)  The  treatment  of  force  and  motion  has  been  consider- 
ably simplified. 

(3)  The  book  has  been  shortened  by  about  sixty  pages,  in 
order  to  give  opportunity  for  an  extended  review  at  the  end  of 
the  course. 

(4)  A  carefully  selected  list  of  review  questions  and  prob- 
lems has  been  inserted  at  the  end. 

(5)  The  absolute  units  have  been  subordinated  even  more 
than  in  the  first  edition ;  for  example,  all  electrical  quantities 
are  defined  in  this  edition  in  terms  of  the  practical,  legal  units. 

(6)  The  presentation  of  the  fundamental  principles  under- 
lying the  dynamo  and  motor  has  been  notably  simplified,  and 
all  of  the  operations  involved  in  these  machines  have  been 
reduced  to  one  simple  rule. 

(7)  The  treatment  of  image  formation  has  been  greatly 
simplified  through  a  more  complete  combination  of  the  wave 
and  ray  methods  than  we  used  before. 

(8)  .More  than  sixty  new  illustrations  and  a  large  number 
of  new  problems  have  been  introduced. 

(9)  New  half-tones  have  been  inserted,  illustrating  some  of 
the  most  notable  achievements  of  modern  physics,  both  in  the 
field  of  application  and  of  pure  science. 


PREFACE  v 

(10)  The  portraits  of  some  of  the  most  eminent  of  modern 
physicists  have  been  inserted,  as  well  as  those  of  the  great 
pioneers  of  the  science. 

The  frontispiece  illustrates  the  combination  of  pure  and 
applied  physics,  which  is  the  guiding  principle  of  the  course. 

For  the  sake  of  indicating  in  what  directions  omissions  may 
be  made  if  necessary,  without  interfering  with  continuity, 
paragraphs  here  and  there  have  been  thrown  into  fine  print. 
These  paragraphs  will  be  easily  distinguished  from  the  class- 
room experiments,  which  are  in  the  same  type.  They  are  for 
the  most  part  descriptions  of  physical  appliances. 

Some  teachers  prefer  to  have  the  chapter  on  heat  trans- 
ference (X)  follow  immediately  after  the  chapter  on  ther- 
mometry  and  expansion  (VII).  This  order  is  as  satisfactory 
to  the  authors  as  that  given. 

It  is  quite  impossible  for  us  to  make  suitable  recognition 
of  the  assistance  which  has  been  derived  from  suggestions 
which  have  been  sent  to  us  from  all  over  the  United  States. 
We  owe  an  especial  debt,  however,  to  H.  Clyde  Krenerick,  of 
Milwaukee  ;  Willard  R.  Pyle,  of  New  York  ;  Willis  E.  Tower 
and  Edwin  S.  Bishop,  of  Chicago. 

THE  UNIVERSITY  OF  CHICAGO  R.  A.  MILLIKAN 

H.  G.  GALE 


CONTENTS 

CHAPTER  PAGE 

I.  MEASUREMENT 1 

Fundamental  Units.    Density 
II.  PRESSURE  IN  LIQUIDS 11 

Liquid  Pressure  beneath  a  Free  Surface.  Pascal's  Law.  The 
Principle  of  Archimedes 

III.  PRESSURE  IN  AIR 26 

Barometric  Phenomena.  Compressibility  and  Expansibility 
of  Air.  Pneumatic  Appliances 

IV.  MOLECULAR  MOTIONS 50 

Kinetic  Theory  of  Gases.  Molecular  Motions  in  Liquids. 
Properties  of  Vapors.  Hygrometry.  Molecular  Motions  in 
Solids 

V.  FORCE  AND  MOTION 74 

Definition  and  Measurement  of  Force.  Composition  and  Reso- 
lution of  Forces.  Gravitation.  Falling  Bodies.  Newton's  Laws 

VI.  MOLECULAR  FORCES 101 

Elasticity.    Capillary  Phenomena.    Absorption  of  Gases 
VII.  THERMOMETRY;  EXPANSION  COEFFICIENTS     ....    116 
Thermometry.    Expansion  Coefficient  of  Gases.    Expansion 
of  Liquids  and  Solids.    Applications  of  Expansion 

VIII.  WORK  AND  MECHANICAL  ENERGY  .* 131 

Definition  arid  Measurement  of  Work.  Work  and  the  Pulley. 
Work  and  the  Lever.  The  Principle  of  Work.  Power  and 
Energy 

IX.  WORK  AND  HEAT  ENERGY       153 

Friction.  Efficiency.  Mechanical  Equivalent  of  Heat.  Specific 
Heat.  Fusion.  Vaporization.  Artificial  Cooling.  Industrial 
Applications 

X.  THE  TRANSFERENCE  OF  HEAT       197 

«• 

Conduction.  Convection.  Radiation.  Heating  and  Ventilating 

vii 


viii  ;  CONTENTS 

CHAPTER  PAGE 

XI.  MAGNETISM 207 

General  Properties  of  Magnets.   Terrestrial  Magnetism 

XII.  STATIC  ELECTRICITY 218 

General  Facts  of  Electrification     Distribution  of  Charge. 
Potential  and  Capacity.    Electrical  Generators 

XIII.  ELECTRICITY  IN  MOTION 240 

Measurement  of  Currents.    Electromotive  Eorce  and  Resist- 
ance.   Primary  Cells 

XIV.  EFFECTS  OF  ELECTRICAL  CURRENTS 267 

Chemical  Effects.    Magnetic  Properties  of  Coils.    Heating 
Effects 

XV.  INDUCED  CURRENTS .    .    284 

The  Principle  of  the  Dynamo  and  Motor.    Dynamos.    The 
Principle  of  the  Induction  Coil  and  Transformer 

XVI.  NATURE  AND  TRANSMISSION  OF  SOUND 314 

Speed  and  Nature.  Reflection,  Reenforcement,  and  Interfer- 
ence 

XVII.  PROPERTIES  OF  MUSICAL  SOUNDS 331 

Musical     Scales.    Vibrating    Strings.     Fundamentals    and 
Overtones.    Wind  Instruments 

XVIII.  NATURE  AND  PROPAGATION  OF  LIGHT 351 

Transmission  of  Light.  The  Nature  of  Light 

XIX.  IMAGE  FORMATION .    .    .    .    . 371 

Images  formed   by   Lenses.    Images   in   Mirrors.    Optical 
Instruments 

XX.  COLOR  PHENOMENA 393 

Color  and  Wave  Length.    Spectra  . 

XXI.  INVISIBLE  RADIATIONS 408 

Radiation  from  a  Hot  Body.  Electrical  Radiations.  Cathode 
and  Rontgen  Rays.    Radioactivity 

APPENDIX      427 

INDEX  .    435 


PORTRAITS  OF  PHYSICISTS  AND  PHOTOGRAPHS 
OF  RECENT  ACHIEVEMENTS   IN  PHYSICS 

PAGE 

1.  Monoplane  flying  over  the  Marconi  .Wireless  Telegraph  Station 
at  Genoa.  2.  C.  T.  R.  Wilson's  Photographs  of  Tracks  of  a-,  /3, 
and  7  Rays,  through  Air Frontispiece 

3.  Archimedes 22 

4.  Otto  von  Guericke       32 

5.  Zeppelin  Airship  Hansa  sailing  over  Hamburg 44 

6.  James  Clerk-Maxwell     ....  * 54 

7.  Heinrich  Rudolph  Hertz 54 

8.  Galileo 88 

9.  Sir  Isaac  Newton 96 

10.  Lord  Kelvin  (Sir  William  Thomson) 122 

11.  James  Watt 148 

12.  James  Prescott  Joule 148 

13.  Count  Rumford  (Benjamin  Thompson) 160 

14.  The  Fisk  Street  Station  of  the  Commonwealth-Edison  Company, 

Chicago,  Illinois 194 

15.  William  Gilbert 218 

16.  Benjamin  Franklin 224 

17.  Count  Alessandro  Volta 234 

18.  Hans  Christian  Oersted 242 

19.  Joseph  Henry 242 

20.  Andre"  Marie  Ampere : 246 

21.  Georg  Simon  Ohm * 252 

22.  Samuel  F.  B.  Morse .276 

23.  Michael  Faraday 284 

24.  Alexander  Graham  Bell 310 

25.  Thomas  A.  Edison 310 

26.  Guglielmo  Marconi 310 

27.  Orville  Wright .  310 

28.  Hermann  von  Helmholtz 340 

29.  Albert  A.  Michelson 352 

30.  Lord  Rayleigh  (John  William  Strutt) 352 

ix 


x  LIST  OF  ILLUSTRATIONS 

PAGE 

31.  Henry  A.  Rowland 352 

32.  Sir  William  Crookes 352 

33.  Christian  Huygens 358 

34.  Arthur  L.  Foley's  Sound- Wave  Photographs 380 

35.  Three-Color  Printing 400 

36.  Cinematograph  Film  of  a  Bullet  fired  through  a  Soap  Bubble    .    .  414 

37.  Joseph  John  Thomson 420 

38.  William  Conrad  Rontgen 426 

39.  Antoine  Henri  Becquerel 426 

40.  Madame  Curie 426 

41.  E.Rutherford  .  426 


,  OF 


A  FIRST  COURSE  IN"  . 
PHYSICS      . 


CHAPTER  I 

MEASUREMENT 
FUNDAMENTAL  UNITS 

1.  Introductory.  A  certain  amount  of  knowledge  about 
familiar  things  comes  to  us  all  very  early  in  life.  We  learn 
almost  unconsciously,  for  example,  that  stones  fall  and  bal- 
loons rise,  that  the  teakettle  stops  boiling  when  removed 
from  the  fire,  that  telephone  messages  travel  by  electric  cur- 
rents, etc.  The  aim  of  physics  is  to  set  us  to  thinking  about 
how  and  why  such  things  happen,  and  to  a  less  degree  to 
acquaint  us  with  other  happenings  which  we  may  not  have 
noticed  or  heard  of  previously.  Most  of  our  accurate  knowl- 
edge about  natural  phenomena  has  been  acquired  by  the 
making  of  careful  measurements.  To  quote  the  words  of 
Lord  Kelvin,  one  of  the,  greatest  physicists  of  the  last  cen- 
tury :  "  When  you  can  measure  what  you»are  speaking  about, 
and  express  it  in  numbers,  you  know  something  about  it; 
and  when  you  cannot  measure  it,  when  you  cannot  express  it 
in  numbers,  your  knowledge  is  of  a  meager  and  unsatisfactory 
kind;  it  may  be  the  beginning  of  knowledge,  but  you  have 
scarcely  in  your  thoughts  advanced  to  the  stage  of  a  science." 

We  can  measure  three  fundamentally  different  kinds  of 
quantities,  —  length,  mass,  and  time,  —  and  we  shall  find  that 
all  other  measurements  may  be  reduced  to  these  three.  Our 

1 


2  f  MEASUREMENT 

first  problem  in  pliysics  is  then  to  learn  something  about 
the  units  in  terms  of  which  all  our  physical  knowledge  is 
expressed. 

2.  Tne    historic    standard   of   length.    Nearly   all  civilized 
nations  have  at  some  time  employed  a  unit  of  length  the 
name  of  which  bore  the   same   significance  as  does  foot  in 
English.    There  can  scarcely  be  any  doubt,  therefore,  that  in 
each  country  this  unit  has  been  derived  from  the  length  of 
the  human  foot.    It  is  probable  that  in  England,  after  the  yard 
(a  unit  which  is  supposed  to  have  represented  the   length 
of  the  arm  of  King  Henry  I)  became  established  as  a  stand- 
ard,  the   foot  was    arbitrarily   chosen   as   one   third   of   this 
standard  yard.    In  view  of  such  an  origin  it  will  be  clear  why 
no   agreement  existed  among  the  units   in   use  in  different 
countries. 

3.  Relations  between  different  units  of  length.    It  has  also 
been  true,  in  general,  that  in  a  given  country  the  different 
units  of  length  in  common  use,  such,  for  example,  as  the 
inch,  the  hand,  the  foot,  the  fathom,  the  rod,  the  mile,  etc., 
have  been  derived  either  from  the  lengths  of  different  mem- 
bers of  the  human  body  or  from  equally  unrelated  magni- 
tudes,  and  in   consequence  have  been   connected  with  one 
another  by  different,  and  often  by  very  awkward,  multipliers. 
Thus,  there  are  12  inches  in  a  foot,  3  feet  in  a  yard,  5J-  yards 
in  a  rod,  1760  yards  in  a  mile,  etc* 

4.  Relations  between  units  of  length,   area,   volume,   and 
mass.   A  similar  and  even  worse  complexity  exists  in  the  rela- 
tions of  the  units  of  length  to  those  of  area,  capacity,  and  mass. 
Thus,  there  are  272^  square  feet  in  a  square  rod ;  57|  cubic 
inches  in  a  quart,  and  31^  gallons  in  a  barrel.    Again  the 
pound,  instead  of  being  the  mass  of  a  cubic  inch  or  a  cubic 
foot  of  water,  or  of  some  other  common  substance,  is  the  mass 
of  a  cylinder  of  platinum,  of  inconvenient  dimensions,  which 
is  preserved  in  London. 


FUNDAMENTAL  UNITS 

5.  Origin  of  the  metric  system.    At  the  time  of  the  French 
Revolution  the  extreme  inconvenience  of  existing  weights  and 
measures,  together  with  the  confusion  arising  from  the  use  of 
different   standards   in  different   localities,  led   the   National 
Assembly  of  France  to  appoint  a  commission  to  devise  a  more 
logical  system.    The  result  of  the  labors  of  this  commission 
was  the  present  metric  system,  which  was  introduced  in  France 
in  1793,  and  has  since  been  adopted  by  the  governments  of 
most  civilized  nations  except  those  of  Great  Britain  and  the 
United  States  ;  and  even  in  these  countries  its  use  in  scientific 
work  is  practically  universal. 

6.  The  standard  meter.    The  standard  length  in  the  metric 
system  is  called  the  meter.    It  is  the  distance,  at  the  freezing 
temperature,  between  two   transverse 

parallel  lines  ruled  on  a  bar  of  platinum- 
iridium  (Fig.  1),  which  is  kept  at  the 
International  Bureau  of  Weights  and 
Measures  at  Sevres,  near  Paris. 

In  order  that  this  standard  length 
might  be  reproduced  if  lost,  the  com- 
mission attempted  to  make  it  one  ten- 
millionth  of  the  distance  from  the 
equator  to  the  north  pole,  measured 
on  the  meridian  of  Paris.  But  since 
later  measurements  have  thrown  some 
doubt  upon  the  exactness  of  the  com- 
mission's determination  of  this  dis- 
tance, we  now  define  the  meter,  not 
as  any  particular  fraction  of  the  earth's 

quadrant,  but  simply  as  the  distance  between  the  scratches 
on  the  bar  mentioned  above.  This  distance  is  39.37  inches, 
or  about  1.1  yards.  On  account  >of  its  more  convenient  size, 
the  centimeter,  one  one-hundredth  of  a  meter,  is  universally 
used  for  scientific  purposes,  as  the  fundamental  unit  of  length. 


FIG.  1.   The  standard 
meter 


r   MEASUREMENT 

7.  Metric  standard  capacity.    The  standard  unit  of  capacity 
is  called  the  liter.    It  is  the  volume  of  a  cube  which  is  one  tenth 
of  a  meter  (about  4  inches)  on  a  side,  and  is  therefore  equal  to 
1000  cubic  centimeters  (cc.).    It  is  equivalent  to  1.057  quarts. 
A  liter  and  a  quart  are  therefore  roughly  equivalent  measures. 

8.  The  metric  standard  of  mass.    In  order  to  establish  a 
connection  between  the  unit  of  length  and  the  unit  of  mass, 
the  commission  directed  a  committee  of  the  French  Academy 
to  prepare  a  cylinder  of  platinum  which  should  have  the  same 
weight  as  a  liter  of  water  at  its  temperature  of  greatest  density, 
namely,  4°  Centigrade  (39°  Fahrenheit).   An  exact  equivalent 
of  this  cylinder  made  of  platinum-iridium,  and  kept  at  Sevres 
with  the  standard  meter,  now  represents  the  standard  of  mass 
in  the  metric  system.    It  is  called  the  standard  kilogram,  and 
is  equivalent  to  about  2.2  pounds.    One  one-thousandth  of  this 
mass  was  adopted  as  the  fundamental  unit  of  mass  and  was 
named  the  gram.    For  practical  purposes,  therefore,  the  gram 
may  be  taken  as  equal  to  the  mass  of  one  cubic  centimeter  of  water. 

9.  The  other  metric  units.    The  three  standard  units  of  the 
metric  system  —  the  meter,  the  liter,  and  the  gram  —  have 
decimal  multiples   and  submultiples,  so  that  every  unit  of 
length,  volume,  or  mass  is  connected  with  the  unit  of  next 
higher  denomination  by  an  invariable  multiplier,  namely,  ten. 

The  names  of  the  multiples  are  obtained  by  adding  the 
Greek  prefixes,  deka  (ten),  hecto  (hundred),  kilo  (thousand); 
while  the  submultiples  are  formed  by  adding  the  Latin  prefixes, 
deci  (tenth),  centi  (hundredth),  and  milli  (thousandth).  Thus : 

1  dekameter  =  10  meters  1  decimeter   =  -^  meter 

1  hectometer  =  100  meters  1  centimeter  =  -^-^  ftieter 

1  kilometer    =  1000  meters  1  millimeter  =  101QO  meter 

The  most  common  of  these  units,  with  the  abbreviations 
which  will  henceforth  be  used  for  them,  are  the  following: 

meter  (m.)  millimeter  (mm.)  gram  (g.) 

kilometer  (km.)          liter  (1.)  kilogram  (kg.) 

centimeter  (cm.)        cubic  centimeter  (cc.)         milligram  (nig.) 


FUNDAMENTAL  UNITS 


10.  Relations  between  the  English  and  metric  units.  The 
following  table  giyes  the  relation  between  the  most  common 
English  and  metric  units. 


1  inch  (in.)  =  ,2.54  cm. 
1  foot  (ft.)  =30.48  cm. 
lmile(mi.)  =  1.609km. 


1  sq. in. 
1  sq.  ft. 

1  cu. in. 
1  cu.  ft. 
Iqt. 


=  6.45  sq.  cm. 
=  929.03  sq.  cm. 

=  16.387  cc. 
=  28,31 7  cc. 
=  .9463  1. 


1  grain  =  64.8  mg. 
loz.  av.  =  28.35  g. 
1  Ib.  av.  =  .4536  kg. 


1  cm.  =  .3937  in. 

1m.  =  1.094yd.  =  39.37  in. 

1km.  =.6214  mi. 

1  sq.  cm.  =  .1550  sq.  in. 

1  sq.  m.  =  1.196  sq.  yd. 

1  cc.  =  .061  cu.  in. 

1  cu.  in.  =  1.308  cu.  yd. 

11.  =1.057qt. 

1  g.  =  15.44  grains 

1  g.  =  .0353  oz. 

1  kg.  =  2.204  Ib. 


This  table  is  inserted  chiefly  for  reference ;  but  the  rela- 
tions 1  in.  =  2.54  cm.,  1m.-  39.37  in.,  1  kilo  (kg.)  =  2.2  Ib., 
1  km.  —  .62  mi.,  should  be  memorized.  Portions  of  a  centimeter 
and  of  an  inch  scale  are  shown  together  in  Fig.  2. 


CENTIMETER 

012345678 

iiiiliiiiliiiiliiiiliinliiii  iiiiliiiiliniliiiilniiliiiiliiiiliiii  iiiilniilnii 


I  I  I  I  I  I  I    I  I  I 

0  INCH 


M    >M    Mi|'M 
123 
FIG.  2.    Centimeter  and  inch  scales 


11.  The  standard  unit  of  time.    The  second  is  taken  among 


86400 


all  civilized  nations  as  the  standard  unit  of  time.    It  is 
part  of  the  time  from  noon  to  noon. 

12.  The  three  fundamental  units.  It  is  evident  that  meas 
urements  of  both  area  and  volume  may  be  reduced  simply  to 
measurements  of  length  ;  for  an  area  is  expressed  as  the  product 
of  two  lengths,  and  a  volume  as  the  product  of  three  lengths. 
For  these  reasons  the  units  of  area  and  volume  are  looked  upon 
as  derived  units,  depending  on  one  fundamental  unit,  the  unit 
of  length. 


6  MEASUREMENT 

Now  it  is  found  that  just  as  measurements  of  area*  and  of 
volume  can  be  reduced  to  measurements  of  length,  so  the  de- 
termination of  any  measurable  quantities,  such  as  the  pressure 
in  a  steam  boiler,  the  velocity  of  a  moving  train,  the  amount 
of  electricity  consumed  by  an  electric  lamp,  the  amount  of 
magnetism  in  a  magnet,  etc.,  can  be  reduced  simply  to  meas- 
urements of  length,  mass,  and  time.  Hence  the  centimeter, 
the  gram,  and  the  second  are  considered  the  three  fundamental 
units.  Whenever  any  measurement  has  been  reduced  to  its 
equivalent  in  terms  of  centimeters,  grams,  and  seconds  it  is 
said,  for  short,  to  be  expressed  in  C.G.S.  (Centimeter-Gram- 
Second)  units. 

13.  Measurement  of  length.    Measuring  the    length   of   a 
body  consists  simply  in  comparing  its  length  with  that  of  the 
standard  meter  bar  kept  at  the  International  Bureau.   In  order 
that  this  may  be  done  conveniently,  great  numbers  of  rods  of 
the  same  length  as  this  standard  meter  bar  have  been  made 
and  scattered  all  over  the  world.  They  are  our  common  meter 
sticks.    They  are  divided  into  10,  100,  or  1000  equal  parts, 
great  care  being  taken  to  have  all  the  parts  of  exactly  the 
same  length.    The  method  of  making  a  measurement  with  such 
a  bar  is  more  or  less  familiar  to  every  one. 

14.  Measurement  of  mass.    Similarly,  measuring  the  mass 
of  a  body  consists  in  comparing  its  mass  with  that  of  the 
standard  kilogram.     In  order  that    this  might  be  done  con- 
veniently, it  was  first  necessary  to   construct  bodies  of  the 
same  mass  as  this  kilogram  and  then  to  make  a  whole  series 
of  bodies  whose  masses  were  |-,  y^-,  TIFo'  TTo~o'  e^c*'  °^  ^e 
mass  of  this  kilogram;  in  other  words,  to  construct  a  set  of 
standard  masses  commonly  called  a  set  of  weights. 

With  the  aid  of  such  a  set  of  standard  masses,  the  deter- 
mination of  the  mass  of  any  unknown  body  is  made  by  first 
placing  the  body  upon  the  pan  A  (Fig.  3)  and  counterpoising 
with  shot,  paper,  etc.,  then  replacing  the  unknown  body  by 


FUNDAMENTAL  UNITS 


3.  The  simple  balance 


as  many  of  the  standard  masses  as  are  required  to  bring  the 
pointer  back  to  0  again.  The  mass  of  the  body  is  equal  to 
the  sum  of  these  standard  masses.  This  is  the  rigorously 
correct  method  of  making  a 
weighing,  and  'is  called  the 
method  of  substitution. 

If  a  balance  is  well  con- 
structed, however,  a  weighing 
may  usually  be  made  with  suffi- 
cient accuracy  by  simply  plac- 
ing the  unknown  body  upon 
one  pan  and  finding  the  sum 
of  the  standard  masses  which 
must  then  be  placed  upon  the 
other  pan  to  bring  the  pointer  again  to  0.  This  is  the  usual 
method  of  weighing.  It  gives  correct  results,  however,  only 
when  the  knife-edge  C  -is  exactly  midway  between  the  points 
of  support  m  and  n  of  the  two  pans.  The  method  of  substitu- 
tion, on  the  other  hand,  is  independent  of  the  position  of  the 
knife-edge. 

QUESTIONS  AND  PROBLEMS 

1.  The  Twentieth  Century  Limited  runs  from  New  York  to  Chicago 
(967  mi.)  in  18  hr.    What  is  its  average  speed  in  miles  per  hour?    in 
kilometers  per  hour? 

2.  Name  as  many  advantages  as  you  can  which  the  metric  system 
has  over  the  English  system.    Can  you  think  of  any  disadvantages  ? 

3.  What  must  you  do  to  find  the  capacity  in  liters  of  a  box  when 
its  length,  breadth,  and  depth  are  given  in  meters  ?  to  find  the  capacity 
in  quarts  when  its  dimensions  are  given  in  feet? 

4.  Find  the  number  of  millimeters  in  5  km.    Find  the  number  of 
inches  in  3  mi.    Which  is  the  easier  ? 

5.  In  the  1912   Gordon  Bennett  aviation  cup  race  Jules  Ve"drines 
in  a  Deperdussin  monoplane  made  a  world's  record  by  flying  20  km.  in 
6  min.  55.9  sec.    What  was  his  speed  in  miles  per  hour  ? 

6.  A  freely  falling  body  starting  from  rest  moves  490  cm.  during 
the  first  second.    Express  this  distance  in  feet. 

t 


8  MEASUREMENT 

DENSITY 

15.  Definition  of  density.  When  equal  volumes  of  different 
substances,  such  as  lead,  wood,  iron,  etc.,  are  weighed  in  the 
manner  described  above,  they  are  found  to  have  widely  differ- 
ent masses.  The  term  "  density  "  is  used  to  denote  the  mass 
of  unit  volume  of  a  substance. 

Thus,  for  example,  in  the  English  system  the  cubic  foot  is 
the  unit  of  volume  and  the  pound  the  unit  of  mass.  Since  1  cubic 
foot  of  water  is  found  to  weigh  62.4  pounds,  we  say  that  in  the  ' 
English  system  the  density  .of  water  is  62 .4  pounds  per  cubic  foot. 

In  the  C.G.S.  system  the  cubic  centimeter  is  taken  as  the 
unit  of  volume  and  the  gram  as  the  unit  of  mass.  Hence  we 
say  that  in  this  system  the  density  of  water  is  1  gram  per 
cubic  centimeter,  for  it  will  be  remembered  that  the  gram  was 
taken  as  the  mass  of  1  cubic  centimeter  of  water.  Unless 
otherwise  expressly  stated,  density  is  now  universally  under- 
stood to  mean  density  in  C.G.S.  units,  that  is,  the  density  of  a 
substance  is  the  mass  in  grams  of  1  cubic  centimeter  of  that  sub- 
stance. For  example,  if  a  block  of  cast  iron  3  cm.  wide,  8  cm. 
long,  and  1  cm.  thick  weighs  177.6  g.,  then,  since  there  are 
24  cc.  in  the  block,  the  mass  of  1  cc.,  that  is,  the  density,  is 
equal  to  1  ^-6,  or  7.4  g.  per  cc. 

The  density  of  some  of  the7  most  common  substances  is 
given  in  the  following  table : 

DENSITIES  OF  SOLIDS 

(In  grams  per  cubic  centimeter) 

Aluminium 2.58  Lead 11.3 

Brass 8.5  Nickel 8.9 

Copper .     8.9  Oak 8 

Cork 24  Pine 5 

Glass 2.6  Platinum 21.5 

Gold 19.3.  Silver    .' 10.53 

Iron  (cast) 7.4  Tin    .    .    . 7.29 

Iron  (wrought) 7.86  Zinc  .    .    .    . 7.15 


DENSITY  9 

DENSITIES  OF  LIQUIDS 

(In  grams  per  cubic  centimeter) 

Alcohol 79  Hydrochloric  acid     .    .    .    .    1.27 

Carbon  bisulphide     ....    1.29  Mercury 13.6 

Glycerin 1.26  Olive  oil 91 

16.  Relation  between  mass,  volume,  and  density.  Since  the 
volume  of  a  body  is  equal  to  the  number  of  cubic  centimeters 
which  it  contains,  and  since  its  density  is  by  definition  the 
number  of  grams  in  1  cubic  centimeter,  its  mass,  that  is,  the  total 
number  of  grams  which  it  contains,  must  evidently  be  equal  to 
its  volume  times  its  density.  Thus,  if  the  density  of  iron  is  7.4 
and  if  the  volume  of  an  iron  body  is  100  cc.,  the  mass  of  this 
body  in  grams  must  equal  7.4  x  100  =  740.  To  express  this 
relation  in  the  form  of  an  equation,  let  M  represent  the  mass 
of  a  body,'  that  is,  its  total  number  of  grams ;  V  its  volume,  that 
is,  its  total  number  of  cubic  centimeters;  and  D  its  density, 
that  is,  the  number  of  grams  in  1  cubic  centimeter;  then 


This  equation  merely  states  the  definition  of  density  in 
algebraic  form. 

17.  Distinction  between  density  and  specific  gravity.  The 
term  "  specific  gravity  "  is  used  to  denote  the  ratio  between  the 
weight  of  a  body  and  the  weight  tf  an  equal  volume  of  water. 
Thus,  if  a  cubic  centimeter  of  iron  weighs  7.4  times  as  much 
as  a  cubic  centimeter  of  water,  its  specific  gravity  is  7.4.  But 
the  density  of  iron  in  C.G.S.  units  is  7.4  grams  per  cubic  cen- 
timeter, for  by  definition  density  in  that  system  is  the  mass 
per  cubic  centimeter.  It  is  clear,  then,  that  density  in  O.  G-.8. 
units  is  numerically  the  same  as  specific  gravity. 

Specific  gravity  is  the  same  in  all  systems,  since  it  simply 
expresses  how  many  times  heavier  a  body  is  than  an  equal  vol- 
ume of  water.  Density,  however,  which  we  have  defined  as  the 
mass  per  unit  volume,  is  different  in  different  systems.  Thus, 


10  MEASUKEMENT 

in  the  English  system  the  density  of  iron  is  461  pounds  per 
cubic  foot  (7.4  x  62.4),  since  we  have  found  that  water  weighs 

62.4  pounds  per  cubic  foot  and  iron  weighs  7.4  times  as  much 
as  an  equal  volume  of  water. 

Since  we  shall  henceforth  use  the  term  "  density  "  to  signify 
exclusively  density  in  the  C.G.S.  system  of  units,  we  shall  have 
little  further  use  in  this  book  for  the  term  "  specific  gravity."  * 

QUESTIONS  AND  PROBLEMS 

1.  If  a  wooden  beam  is  30  X  20  x  500  cm.  and  has  a  mass  of  150  kg., 
what  is  the  density  of  wood  ? 

2.  Would  you  attempt  to  carry  home  a  block  of  gold  the  size  of  a 
peck  measure?    (Consider  a  peck  equal  to  8  1.    See  table,  p.  8.) 

3.  What  is  the  mass  of  a  liter  of  alcohol? 

4.  How  many  cubic  centimeters  in  a  block  of  brass  weighing  34  g.  ? 

5.  What  is  the  weight  in  metric  tons  of  a  cube  of  lead  2  m.  on  an 
edge  ?    (A  metric  ton  is  1000  kilos,  or  about  2200  Ib.) 

6.  Find   the  volume  in  liters  of  a  block  of  platinum  weighing 

45.5  kilos. 

— — •  7.  Find  the  density  of  a  steel  sphere  of  radius  1  cm.  and  weight 
32.7  g. 

8.  One  kilogram  of  alcohol  is  poured  into  a  cylindrical  vessel  and 
fills  it  to  a  depth  of  8  cm.    Find  the  cross  section  of  the  cylinder. 

9.  A  capillary  glass  tube  weighs  .2  g'.    A  thread  of  mercury  10  cm. 
long  is  drawn  into  the  tube,  when  it  is  found  to  weigh  .6  g.    Find  the 
cross  section  of  the  capillary  tube". 

10.  Find  the  length  of  a  lead  rod  1  cm.  in  diameter  and  weighing  1  kg. 

*  Laboratory  exercises  on  length,  mass,  and  density  measurements  should 
accompany  or  follow  this  chapter.  See,  for  example,  Experiments  1,  2,  and  3  of 
the  authors'  manual. 


CHAPTER  II 

PRESSURE  IN  LIQUIDS 

LIQUID  PRESSURE  BENEATH  A  FREE  SURFACE 

18.  Force  beneath  the  surface  of  a  liquid.  We  are  all  con- 
scious of  the  'fact  that  in  order  to  lift  a  kilogram  of  mass  we 
must  exert  an  upward  pull.  Experience  has  taught  us  that 
the  greater  the  mass,  the  greater  the  force  which  we  must 
exert.  In  fact,  the  force  is  commonly  taken  as  numerically 
equal  to  the  mass  lifted.  This  is  calledv  the  weight  measure  of 
a  force.  A  push  or  pull  ivhich  is  equal  to  that  required  to  sup- 
port a  gram  of  mass  is  catted  a  gram  of  force. 

To  investigate  the  nature  of  the  forces  beneath  the  free  surface  of 
a  liquid,  we  shall  use  a  pressure  gauge  of  the  form  shown  in  Fig.  4.  If 
the  rubber  diaphragm  which  is  stretched  across  the  mouth  of  a  thistle 
tube  A  is  pressed  in  lightly  with  the  finger,  the  drop  of  ink  B  will  be 
observed  to  move  forward  in  the  tube  T,  but  it  will  return  again  to 
its  first  position  as  soon  as  the  finger  is  removed.  If  the  pressure  of  the 
finger  is  increased,  the  drop  will  move  forward  a  greater  distance  than 
before.  We  may  therefore  take  the  amount  of  motion  of  the  drop  as  a 
measure  of  the  amount  of  force  acting  on  the  diaphragm. 

Now  let  A  be  pushed  down  first  2,  then  4,  then  8  cm.  below  the  sur- 
face of  the  water  (Fig.  4).  The  motion  of  the  index  B  will  show  that 
the  upward  force  continually  increases  as  the  depth  increases. 

Careful  measurements  made  in  the  laboratory  will  show 
that  the  force  is  directly  proportional  to  the  depth.* 

*  It  is  recommended  that  quantitative  laboratory  work  on  the  law  of  depths 
and  on  the  use  of  manometers  accompany  this  discussion.  See,  for  example,  Experi- 
ments 4  and  5  of  the  authors'  manual. 

11 


12  PRESSURE  IN  LIQUIDS 

Let  the  diaphragm  A  (Fig.  4)  be  pushed  down  to  some  convenient 
depth,  for  example,  10  centimeters,  and  the  position  of  the  index  noted. 
Then  let  it  be  turned  sidewise  _. 

so  that  its  plane  is  vertical 
(see  a,  Fig.  4),  and  adjusted  in 
position  until  its  center  is  ex-  f^Li—  jf^.,  CJL 


actly  10  centimeters  beneath 

J  FIG.  4.    Gauge  for  measuring  liquid 

the  surface,  that  is,  until  the  pressure  ' 

average    depth    of    the    dia- 

phragm is  the  same  as  before.    The  position  of  the  index  will  show  that 
the  force  is  also  exactly  the  same  as  before. 

Let  the  diaphragm  then  be  turned  to  the  position  ft,  so  that  the  gauge 
measures  the  downward  force  at  a  depth  of  10  centimeters.  The  index 
will  show  that  this  force  is  again  the  same. 

We  conclude,  therefore,  that  at  a  given  depth  a  liquid  presses 
up  and  down  and  sidewise  on  a  given  surface  with  exactly  the 
same  force. 

19.  Magnitude  of  the  force.  If  a  vessel  like  that  shown  in 
Fig.  5  is  filled  with  a  liquid,  the  force  against  the  bottom  is 
obviously  equal  to  the  weight  of  the  column  of  liquid  resting 
upon  the  bottom.  Thus,  if  F  represents  this  force  in  grams,  A 
the  area  in  square  centimeters,  h  the  depth  in  centimeters, 
and  d  the  density  in  grams  per  cubic  centimeter,  we 
shall  have 

F=Ahd.  (1) 


Since,  as  was  shown  by  the  experiment  of  the  preced- 
ing section,  the  force  is  the  same  in  all  directions  at 
a  given  depth,  we  have  the  following  general  rule  : 

The  force  which  a  liquid  exerts  against  any  surface  is  equal 
to  the  area  of  the  surface  times  its  average  depth  times  the  density 
of  the  liquid. 

It  is  important  to  remember  that  "average  depth"  means 
the  vertical  distance  from  the  level  of  the  free  surface  to  the 
center  of  the  area  in  question. 


PRESSURE  BENEATH  A  FREE  SURFACE 


13 


20.  Pressure  in  liquids.    Thus  far  attention  has  been  con- 
fined to  the  total  force  exerted  by  a  liquid  against  the  whole 
of  a  given  surface.    It  is  often  more  convenient  to  consider 
the  surface  divided  into  square  centimeters  or  square  inches, 
and  consider  the  force  on   one  of  these  units   of  area.    In 
physics  the  word  "  pressure  "  is  used  exclusively  to  denote 
the  force  per  unit  area.    Pressure  is  thus  a  measure  of  the 
intensity  of  the  force  acting  on  a  surface,  and  does  not  depend 
at  all  on  the  area  of  the  surface.   Since,  by  §  19,  F=  Ahd,  and 
since  by  definition  the  pressure  p  is  equal  to  the  force  per 
unit  area,  we  have 

p  =  —  =  hd.  (2) 

Therefore  t he  pressure  at  a  depth  of  h  centimeters  below  the 
surface  of  a  liquid  of  density  d  is  hd  grams  per  square  centimeter. 

If  the  height  is  given  in  feet  and  the  density  in  pounds  per 
cubic  foot,  namely,  62.4,  then  the  product  hd  gives  pressure 
in  pounds  per  square  foot.  Dividing  by  144  gives  the  result 
in  pounds  per  square  inch. 

21.  Levels  of  liquids  in  connecting  vessels.    It  is  a  perfectly 
familiar  fact  that  when  water  is  poured  into  a  teapot  it  stands 
at   exactly   the    same    level '  in    the 

spout  as  in  the  body  of  the  teapot ; 
or  if  it  is  poured  into  a  number  of 
connected  tubes,  like  those  shown  in 
Fig.  6,  the  surfaces  of  the  liquid  in 
the  various  tubes  lie  in  the  same 
horizontal  plane.  Now  the  pressure 
at  <?,  Fig.  7,  was  shown  by  the  experi- 
ment of  §  18  to  be  equal  to  the  den- 
sity of  the  liquid  times  the  depth  eg. 
The  pressure  at  o  in  the  opposite  direction  must  be  equal  to 
that  at  <?,  since  the  liquid  does  not  tend  to  move  in  either  direc- 
tion. Hence  the  pressure  at  o  must  be  ks  times  the  density. 


FIG.  6.    Water  level  in  com- 
municating vessels 


14 


PKESSURE  IN  LIQUIDS 


If  water  is  poured  in  at  *  so  that  the  height  k*  is  increased, 
the  pressure  to  the  left  at  o  becomes  greater  than  the  pres- 
sure to  the  right  at  <?,  and  a  flow  of  water  takes  place  to  the 
left  until  the  heights  are  again  equal. 

It  follows  from  these  observations 
on  the  level  of  water  in  connected  ves- 
sels that  the  pressure  beneath  the  sur- 
face of  a  liquid  depends  simply  on  the 
vertical  depth  beneath  the  free  surface, 
and  not  at  all  on  the  size  or  shape  of 
the  vessel. 


FIG.  7.   Why  water  seeks 
its  level 


FIG. '8.    Illustrating  hydro- 
static paradox 


QUESTIONS  AND  PROBLEMS 

1.  If  the  areas  of  the  surfaces  AB  in  Fig.  8,  (1)  and  (2)  are  the 
same,  and  if  water  is  poured  into  each  vessel  at  D  till  it  stands  at  the 
same  height  above  AB,  how  will  the  downward  force  on  AB  in  Fig.  8,  (2) 
compare  with  that  in  Fig.  8,  (1)?    Test 

your  answer,  if  possible,  by  making  AB  a 
piece  of  cardboard  and  pouring  water  in 
at  D  in  each  case  until  the  cardboard 
is  forced  off. 

2.  Soundings  at  sea  are  made  by  low- 
ering  some    kind  of   a    pressure    gauge. 
When  this  gauge  reads  1.3  kg.  per  square 
centimeter,  what  is  the  depth  ?  (Density 
of  sea  water  =  1.026.) 

3.  If  the  pressure  at  a  tap  on  the  first  floor  reads  80  Ib.  per  square 
inch,  and  at  a  tap  two  floors  above  68  Ib.,  what  is  the  difference  in  feet 
between  the  levels  of  the  two  taps  ? 

4.  If  the  vessel  shown  in  Fig.  10,  (1),  p.  15,  has  a  base  of  200  sq.  crn. 
and  if  the  water  stands  100  cm.  deep,  what  is  the  total  force  on  the 
bottom  ? 

5.  If  the  weight  of  the  empty  vessel  in  Fig.  10,  (1)  is  small  compared 
with  the  weight  of  the  contained  water,  will  the  force  required  to  lift  the 
vessel  and  water  be  greater  or  less  than  the  force  exerted  by  the  water 
against,  the  bottom  ?   Explain. 

6.  Find  the  total  force  against  the  gate  of  a  lock  if  its  width  is  60  ft. 
and  the  depth  of  the  water  20  ft.   Will  it  have  to  be  made  stronger  if  it 
holds  back  a  lake  than  if  it  holds  back  a  small  pond  ? 


PASCAL'S  LAW 


15 


7.  A  whale  when   struck  with  a  harpoon  will  often  dive  straight 
down  as  much  as  400  fathoms  (2400  ft.).   If  the  body  is  60  ft.  long  and 
has  an  average  circumference  of  15  ft.,  what  is  the  total 

force  to  which  it  is  subjected? 

8.  A  hole  5  cm.  square  is  made  in  a  ship's  bottom 
7  m.  below  the  water  line.    What  force  in  kilograms  is 
required  to  hold  a  board  above  the  hole  ? 

9.  Thirty    years    ago    standpipes    were    generally 
straight  cylinders.    To-day  they  are  more  commonly 
of  the  form  shown  in  Fig.  9.   What  are  the  advantages 
of  each  form  ? 

PASCAL'S  LAW 

22.  Transmission    of    pressure    by    liquids. 
From    the    fact    that   pressure    within    a   free 
liquid   depends    simply   upon   the   derjtlj   and 
density  of  the  liquid,  it  is  possible  to  deduce     FIG.  9.  A  water 
a  very  surprising  conclusion,  which  was  first 
stated  by  the  famous  French  scientist,  mathematician,  and 
philosopher,  Pascal  (1623-1662). 

Let  us  imagine  a  vessel  of  the  shape  shown  in  Fig.  10,  (1), 
to  be  filled  with  water  up  to  the  level  ah.  For  simplicity  let 
the  upper  portion  be  assumed  to 
be  1  square  centimeter  in  cross 
section.  Since -the  density  of  water 
is  1,  the  force  with  which  it  presses 
against  any  square  centimeter  of 
the  interior  surface  which  is  h  cen- 
timeters beneath  the  level  ab  is  h 
grams.  Now  let  1  gram  of  water 
(that  is,  1  cubic  centimeter)  be 
poured  into  the  tube.  Since  each 

square  centimeter  of  surface  which 

,       FIG.  10.   Proof  of  Pascal's  law 
was  before  h  centimeters  beneath 

the  level  of  the  water  in  the  tube  is  now  h  4-  1  centimeters 
beneath  this  level,  the  new  pressure  which  the  water  exerts 


a 

fl) 

*• 

b 

^1- 

IH{S=^ 

L    —  —    — 

-j-_—  —  ~p-=" 

i 

16  PKESSUKE  IN  LIQUIDS 

against  it  is  h  + 1  grams ;  that  is,  applying  1  gram  of  force  to 
the  square  centimeter  of  surface  ab  has  added  1  gram  to  the 
force  exerted  by  the  liquid  against  each  square  centimeter  of 
the  interior  of  the  vessel.  Obviously  it  can  make  no  differ- 
ence whether  the  pressure  which  was  applied  to  the  surface 
ab  was  due  to  a  weight  of  water  or  to  a  piston  carrying  a 
load,  as  in  Fig.  10,  (2),  or  to  any  other  cause  whatever.*  We 
thus  arrive  at  Pascal's  conclusion  that  pressure  applied  any- 
where to  a  body  of  confined  liquid  is  transmitted  undiminished 
to  every  portion  of  the  surface  of  the  containing* vessel. 

23.  Multiplication  of  force  by  the  transmission  of  pressure 
by  liquids.    Pascal  himself  pointed  out  that  with  the  aid  of 
the  principle  stated  above  we  ought  to  be  able  to  transform 
a  very  small  force  into  one  of  un- 
limited    magnitude.     Thus,    if    the 
area  of  the  cylinder  ab  (Fig.  11)  is 
1  sq.  cm.,  while  that  of  the  cylinder 
AB  is  1000  sq.  cm.,  a  force  of  1  kg.     FlG<  1L  Multiplication  of 
applied  to  ab  would  be  transmitted     force   by   transmission   of 
by  the  liquid  so  as  to  act  with   a  pressure 

force  of  1  kg.  on  each,  square  centimeter  of  the  surface  AB. 
Hence  the  total  upward  force  exerted  against  the  piston  AB 
by  the  1  kg.  applied  at  ab  would  be  1000  kg.  Pascal's  own 
words  are  as  follows:  "  A  vessel  full  of  water  is  a  new  prin- 
ciple in  mechanics,  and  a  new  machine  for  the  multiplication 
of  force  to  any  required  extent,  since  one  man  will  by  this 
means  be  able  to  move  any  given  weight." 


24.  The  hydraulic  press.  The  experimental  proof  of  the  correctness 
of  the  conclusions  of  the  preceding  paragraph  is  furnished  by  the 
hydraulic  press,  an  instrument  now  in  common  use  for  subjecting  to 
enormous  pressures  paper,  cotton,  etc.,  and  for  punching  holes  through 
iron  plates,  testing  the  strength  of  iron  beams,  extracting  oil  from 
seeds,  making  dies,  embossing  metal,  etc. 


PASCAL'S  LAW 


17 


Such  a  press  is  represented  in  section  in  Fig.  12.  As  the  small  piston  p 
is  raised,  water  from  the  cistern  C  enters  the  piston  chamber  through  the 
valve  v.  As  soon  as  the  down  stroke  begins,  the  valve  v  closes,  the  valve 
?/  opens,  and  the  pressure  applied  on  the  piston  p  is  transmitted  through 


FIG.  12.    Diagram  of  a  hydraulic  press 

the  tube  K  to  the  large  reservoir,  where  it  acts  on  the  large  cylinder  P 
with  a  force  which  is  as  many  times  that  applied  to  p  as  the  area  of  P 
is  times  the  area  of  p. 

Hand  presses  similar  to  that  shown  in.  Fig.  12  are  often  made  which 
are  capable  of  exerting  a  compressing  force  of  from  500  to  1000  tons. 

25.  No  gain  in  the  product  of  force  times  distance.  It  should 
be  noticed  that,  while  the  force  acting  on  AB  (Fig.  11)  is 
1000  times  as  great  as  the  force  acting  on  a£,  the  distance 


18 


PRESSUKE  IN  LIQUIDS 


through  which  the  piston  AB  is  pushed  up  in  a  given  time  is 
of  the  distance  through  which  the  piston  ab  moves 


1000 


but 

down.  For,  forcing  ab  down  a  distance  of  1  centimeter  crowds 
but  1  cubic  centimeter  of  water  over  into  the  large  cylin- 
der, and  this  additional  cubic  centimeter  can  raise  the  level 
of  the  water  there 


but- 


-centimeter. 


1000 

We  see,  therefore, 
that  the  product  of 
the  force  acting  by 
the  distance  moved 
is  precisely  the  same 
at  both  ends  of  the 
machine.  This  im- 
portant conclusion 
will  be  found  in 
our  future  study 
to  apply  to  all  ma- 
chines. 

26.  The  hydraulic 
elevator.  Another  very 
common  application  of 
the  principle  of  trans- 
formation of  pressure 
by  liquids  is  found  in 
the  hydraulic  elevator. 
The  simplest  form  of 
such  an  elevator  is 
shown  in  Fig.  13.  The 
cage  A  is  borne  on 
the  top  of  a  long  piston  P  which  runs  in  a  cylindrical  pit  C  of  the 
same  depth  as  the  height  to  which  the  carriage  must  ascend.  Water 
enters  the  pit  either  directly  from  the  water  mains  m  of  the  city's 
supply,  or,  if  this  does  not  furnish  sufficient  pressure,  from  a  special 
reservoir  on  top  of  the  building.  When  the  elevator  boy  pulls  up  on  the 
cord  cc,  the  valve  v  opens  so  as  to  make  connection  from  m  into  C.  The 


FIG.  13  FIG.  14 

Diagrams  of  hydraulic  elevators 


PASCAL'S  LAW 


19 


elevator  then  ascends.    When  cc  is  pulled  down,  v  turns  so  as  to  permit 
the  water  in  C  to  escape  into  the  sewer.    The  elevator  then  descends. 

Where  speed  is  required  the  motion  of  the  cylinder  is  communicated 
indirectly  to  the  cage  by  a  system  of  pulleys  like  that  shown  in  Fig.  14. 
With  this  arrangement  a  foot  of  upward  motion  of  the  cylinder  P 
causes  the  counterpoise  D  of  the  cage  to  descend  2  feet,  for  it  is  clear 
from  the  figure  that  when  the  cylinder  goes  up  1  foot  enough  rope  must 
be  pulled  over  the.  fixed  pulley  p  to  lengthen  each  of  the  two  strands  a 
and  b  1  foot.  Similarly,  when  the  counterpoise  descends  2  feet  the  cage 
ascends  4  feet.  Hence  the  cage  moves  four  times  as  fast  and  four  times 
as  far  as  the  cylinder.  The  elevators  in  the  Eiffel  Tower  in  Paris  are 
of  this  sort.  They  have  a  total  travel  of  420  feet  and  are  capable  of 
lifting  50  people  400  feet  per  minute. 

27.  City  water  supply.  Fig.  15  illustrates  the  method  by 
which  a  city  is  often  supplied  with  water  from  a  distant  source. 
The  aqueduct  from  the  lake  a  passes  under  a  road  r,  a  brook 
b,  a  hill  H,  and  into  a  reservoir  e,  from  which  it  is  forced  by 
the  pump  p  into  the  standpipe  P,  whence  it  is  distributed  to 
the  houses  of  the  city.  If  a  static  condition  prevailed  in  the 
whole  system,  then  the  water  level  in  e  would  of  necessity  be 


FIG.  15.    City  water  supply  from  lake 

the  same  as  that  in  a,  and  the  level  in  the  pipes  of  the  building 
B  would  be  the  same  as  that  in  the  standpipe  P.  But  when 
the  water  is  flowing,  the  friction  of  the  mains  causes  the  level 
in  e  to  be  somewhat  less  than  that  in  #,  and  that  in  B  less 
than  that  in  P.  It  is  on  account  of  the  friction  both  of  the 
air  and  of  the  pipes  that  the  fountain  f  does  not  actually  rise 
nearly  as  high  as  the  ideal  limit  shown  in  the  figure  (see 
dotted  line). 


20  PRESSURE  IN  LIQUIDS 

28.  Artesian  wells.  It  is  in  the  principle  of  transmission  of  pressure 
by  liquids  that  artesian  wells  find  their  explanation.  Fig.  16  is  an  ideal 
section  of  what  geologists  call  an  artesian  basin.  The  stratum  A  is 
composed  of  some  porous  material  such  as  sand,  open-textured  sand- 
stone, or  broken  rock,  through  which  the  water  can  percolate  easily. 
Above  and  below  it  are  strata  C  and  B  of  clay,  slate,  or  some  other 
material  impervious  to  water.  The  porous  layer  is  filled  with  water 
which  finds  entrance  at  the  outcropping  margins.  As  soon  as  a  boring 


FIG.  16.    Artesian  wells 

is  made  through  the  layer  C  the  water  gushes  forth  because  of  the  trans- 
mission of  pressure  from  the  higher  levels.  A  well  of  this  sort  exists 
near  Leipzig,  Germany,  which  is  5735  feet  deep.  Many  artesian  wells 
have  been  bored  in  the  desert  of  Sahara  and  an  abundant  water  supply 
found  at  a  depth  of  200  feet.  Great  numbers  of  these  wells  exist  in 
the  United  States,  notable  ones  being  located  at  Chicago,  Louisville 
(Kentucky),  and  Charleston  (South  Carolina).  The  artesian  basins  in 
which  the  wells  are  found  are  often  a  hundred  miles  or  more  in  width. 


QUESTIONS  AND  PROBLEMS 

1.  How  does  your  city  get  its  water?    How  is  the  pressure  in  the 
pipes  maintained? 

2.  A  jug  full  of  water  may  often  be  burst  by  striking  a  blow  on 
the  cork.    If  the  surface  of  the  jug  is  200  sq.  in.  and  the  cross  section  of 
the  cork  1  sq.  in.,  what  total  force  acts  on  the  interior  of  the  jug  when 
a  10-lb.  blow  is  struck  on  the  cork?    /    to  ^ 

3.  If  the  water  pressure  in  the  city  mains  is  70  Ib.  to  the  square  inch, 
how  high  above  the  town  is  the  top  of  the  water  in  the  standpipe  ? 

4.  A  cubical  box  10  cm.  on  a  side  is  half  filled  with  mercury  and 
half  with  water.    Find  the  total  force  in  grams  on  the  bottom ;  on 
each  side.  <2  "  7 

5.  The  water  pressure  in  the  city  mains  is  80  Ib.  to  the  square  inch. 
The  diameter  of  the  piston  of  a  hydraulic  elevator  of  the  type  shown 
in  Fig.  13  is  10  in.    If  friction  could  be  disregarded,  how  heavy  a  load 
could  the  elevator  lift  ?   If  30  %  of  the  ideal  value  must  be  allowed  for 
frictional  loss,  what  load  will  the  elevator  lift? 


THE  PRINCIPLE  OF  ARCHIMEDES 


21 


17  17 

FIG.  17 

Hydrostatic 
bellows 


6.  Fig.  17  represents  an  instrument  commonly  known  as  the  hydro- 
static bellows.   If  the  base  C  is  20  in.  square  and  the  tube  is  filled  with 
water  to  a  depth  of  5  ft.  above  the  top  of  C,  what  is  the 

value  of  the  weight  which  the  bellows  can  support  ? 

7.  A  hydraulic  press  having  a  piston  1  in.  in  diameter 
exerts  a  force  of  10,000  Ib.  when  10  Ib.  are  applied  to  this 
piston.    What  is  the  diameter  of  the  large  piston  ? 


THE  PRINCIPLE  OF  ARCHIMEDES  * 

29.  Loss  of  weight  of  a  body  in  a  liquid.  The 
preceding  experiments  have  shown  that  an  up- 
ward force  acts  against  the  bottom  of  any  body 
immersed  in  a  liquid.  If  the  body  is  a  boat,  cork, 
piece  of  wood,  or  any  body  which  floats,  it  is 
clear  that,  since  it  is  in  equilibrium,  this  upward 
•force  must  be  equal  to  the  weight  of  the  body. 
Even  if  the  body  does  not  float,  everyday  obser- 
vation shows  that  it  still  loses  a  portion  of  its  natural  weight, 
for  it  is  well  known  that  it  is  easier  to  lift  a  stone  under 
water  than  in  air;  or,  again,  that  a  man  in  a  bath  tub  can 
support  his  whole  weight  by  pressing  'lightly  against  the 
bottom  with  his  fingers.  It  was  indeed  this  very  observation 
which  first  led  the  old  Greek  philosopher  Archimedes  (287- 
212  B.C.)  to  the  discovery  of  the  exact  law  which  governs 
the  loss  of  weight  of  a  body  in  a  liquid. 

Hiero,  the  tyrant  of  Syracuse,  had  ordered  a  gold  crown 
made,  but  suspected  that  the  artisan  had  fraudulently  used 
silver  as  well  as  gold  in  its  construction.  He  ordered  Archi- 
medes to  discover  whether  or  not  this  were  true.  How  to  do 
so  without  destroying  the  crown  was  at  first  a  puzzle  to  the 
old  philosopher.  While  in  his  daily  bath,  noticing  the  loss  of 
weight  of  his  own  body,  it  suddenly  occurred  to  him  that  any 

*  A  laboratory  exercise  on  the  experimental  proof  of  Archimedes'  principle 
should  precede  or  accompany  this  discussion.  See,  for  example,  Experiment  6  of 
the  authors'  manual. 


22 


PRESSURE  IN  LIQUIDS 


, 


FIG.  18.  Proof  that 
an  immersed  body 
is  buoyed  up  by  a 
force  equal  to  the 
weight  of  the  dis- 
placed liquid 


body  immersed  in  a  liquid  must  lose  a  weight  equal  to  the  weight 
of  thet  displaced  liquid.  He  is  said  to  have  jumped  afr  once  to 
his  feet  and  rushed  through  the  streets  of  Syracuse  crying, 
"  Eureka,  eureka !  "  (I  have  found  it,  I  have  found  it !) 

30.  Theoretical  proof  of  Archimedes'  principle.  It  is  probable 
that  Archimedes,  with  that  faculty  which  is  so  common  among 
men  of  great  genius,  saw  the  truth  of  his 
conclusion  without  going  through  any  log- 
ical process  of  proof.  Such  a  proof,  how- 
ever, can  easily  be  given.  Thus,  since  the 
upward  force  on  the  bottom  of  the  block 
abed  (Fig.  18)  is  equal  to  the  weight  of  the 
column  of  liquid  obce,  and  since  the  down- 
ward force  on  the  top  of  this  block  is  equal 
to  the  weight  of  the  column  of  liquid  oade, 
it  is  clear  that  the  upward  force  must  ex- 
ceed the  downward  force  by  the  weight  of 
the  column  of  liquid  abccf;  that  is,  the  buoyant  force  exerted  by 
the  liquid  is  exactly  equal  to  the  weight  of  the  displaced  liquid. 

The  reasoning  is^  exactly  the  same,  no 
matter  what  may  be  the  nature  of  the 
liquid  in  which  the  body  is  immersed,  nor 
how  far  the  body  may  be  beneath  the  sur- 
face. Further,  if  the  body  weighs  more 
than  the  liquid  which  it  displaces,  it  must 
sink,  for  it  is  urged  down  with  the  force 
of  its  own  weight,  and' up  with  the  lesser 
force  of  the  weight  of  the  displaced  liquid. 
But  if  it  weighs  less  than  the  displaced 
liquid,  then  the  upward  force  due  to  the 
displaced  liquid  is  greater  than  its  own  weight,  and  con- 
sequently it  must  rise  to  the  surface.  When  it  reaches  the 
surface  the  downward  force  on  the  top  of  the  block,  due  to 
the  liquid,  becomes  zero.  The  body  must,  however,  continue 


m 


FIG.  19.  Proof  that 
a  floating  body  is 
buoyed  up  by  a 
force  equal  to  the 
weight  of  the  dis- 
placed liquid 


Archimedes  (287-212  B.C.) 
(Bust  in  Naples  Museum) 

The  celebrated  geometrician  of  antiquity;  lived  at  Syracuse, 
Sicily;  first  made  a  determination  of  IT  and  computed  the  area 
of  the  circle ;  discovered  the  laws  of  the  lever  and  was  author  of 
the  famous  saying,  "  Give  me  where  I  may  stand  and  I  will  move 
the  world";  discovered  the  laws  of  flotation;  invented  various 
devices  for  repelling  the  attacks  of  the  Romans  in  the  siege  of 
Syracuse ;  on  the  capture  of  the  city,  while  in  the  act  of  drawing 
geometrical  figures  in  a  dish -of  sand,  he  was  killed  by  a  Roman 
soldier  to  whom  he  cried  out,  "Don't  spoil  my  circle." 


THE  PRINCIPLE  OF  ARCHIMEDES  23 

to  rise  until  the  upward  force  on  its  bottom  is  equal  to  its 
own  weight.  But  this  upward  force  is  always  equal  to  the 
weight  of  the  displaced  liquid,  that  is,  to  the  weight  of  the 
column  of  liquid  mbcn  (Fig.  19). 

Hence  a  floating  body  must  displace  its  own  iveight  of  the 
liquid  in  .which  it  floats.  This  statement  is  embraced  in  the 
original  statement  of  Archimedes'  principle,  for  a  body  which 
floats  has  lost  its  whole  weight. 

31.  Density  of  a  heavy  solid.    The  density  of  a  body  is,  by 
definition,  its  mass  divided  by  its  volume.    It  is  always  pos- 
sible to  obtain  the  mass  of  a  body  by  weighing  it,  but  it  is 
not,  in  general,  possible  to  obtain  the  volume 
of  an  irregular  body  from  measurements  of 
its  dimensions.    Archimedes'  principle,  how- 
ever, furnishes  an  accurate  and  easy  method 
for  obtaining  the  volume  ,of  any  solid,  how- 
ever irregular,  for  by  the  preceding  paragraph 
this  volume  is  numerically  equal  to  the  loss 
of  weight  'in  water.     Hence    the    equation 
which  defines  density,  namely, 

Density  = ^L__  FIG.  20.  Method  of 

Volume  ^  weighing    a    body 

becomes  in  this  case  under  water 

T>.       ..  Mass 

Density  =  —  —  •  (3) 

Loss  of  weight  in  water 

Fig.  20  shows  the  usual  arrangement  for  finding  the  weight 
in  water. 

3£.  Density  of  a  solid  lighter  than  water.  If  the  body  is 
too  light  to  sink  of  itself,  we  may  still  obtain  its  volume  by 
forcing  it  beneath  the  surface  with  a  sinker.  Thus,  suppose 
wl  represents  the  weight  on  the  right  pan  of  the  balance  when 
I  the  body  is  in  air  and  the  sinker  in  water,  as  in  Fig.  21 ;  while 

t 


24 


PBESSUEE  IN  LIQUIDS 


FIG.  21.    Method  of  finding  density 
of  a  light  solid 


w2  is  the  weight  on  the  right  pan  when  both  body  and  sinker 
are  under  water.  Then  w1  —  w2  is  obviously  the  buoyant  effect 
of  the  water  on  the  body 
alone,  and  is  therefore  equal 
to  the  weight  of  the  dis- 
placed water,  which  is  numer- 
ically equal  to  the  volume 
of  the  body. 

33.  Density  of  liquids  by 
the  hydrometer  method.  The 
commercial  hydrometer  such 
as  is  now  in  common  use  for 
testing  the  density  of  alco- 
hol, milk,  acids,  sugar  solu- 
tions,   etc.    is    of    the    form 
shown  in  Fig.  22.    The  stem 

is  calibrated  by  trial  so  that  the  density  of  any  liquid  may  be 
read  upon  it  directly.  The  principle  involved  is  that  a  float- 
ing body  sinks  until  it  displaces  its  own  weight. 
By  making  the  stem  very  slender  the  sensi- 
tiveness of  the  instrument  may  be  made  very 
great.  Why? 

34.  Density  of  liquids,  by  "  loss  of  weight" 
method.    If  any  suitable  solid  be  weighed,  first 
in  air,  then  in  water,  and  then  in  some  liquid 
of  some  unknown  density,  by  the  principle  of 
Archimedes,  the  loss  of  weight  in  the  liquid  is 
equal  to  the  weight  of  the   liquid  displaced, 
and  the  loss  in  water  is  equal  to  the  weight  of 

the  water  displaced.    Obviously  the  volume  of  FlG-  22- 

liquid  displaced  is  the  same,  since  in  each  case  it 

is  just  the  volume  of  the  body.    If  we  divide  the 

loss  of  weight  in  the  liquid  by  the  loss  of  weight  in  water, 

we  are  dividing  the  weight  of  a  given  volume  of  liquid  by 


THE  PRIHCIPLE  OE  ARCHIMEDES  25 

the  weight  of  an  equal  volume  of  water.  By  §  17  this  gives 
us  at  once  the  specific  gravity  of  the  liquid,  which  is  the  same 
as  its  density  in  the  C.G.S.  system.  Therefore,  to  find  the 
density  of  a  liquid,  divide  the  loss  of  weight  of  some  solid  in 
it  ly  the  loss  of  weight  of  the  same  body  in  water.* 

QUESTIONS  AND  PROBLEMS 

1.  What  fraction  of  the  volume  of  a  block  of  wood  will  float  above 
water  if  its  deusity  is  .5  ?  if  its  density  is  .6  ?  if  its  density  is  .9  ?    State 
in  general  what  fraction  of  the  volume  of  a  floating  body  is  under  water. 

2.  If  an  iceberg  rises  100  ft.  above  water,  how  far  does  it  extend 
below  water?    (Assume  the  density  of  the  ice  to  be  .9   that  of   sea 
water.) 

3.  Tf  a  barge  30  ft.  by  15  ft.  sank  4  in.  when  an  elephant  was  taken 
aboard,  what  was  the  elephant's  weight? 

4.  The  hull  of  a  modern  battleship  is  made  almost  entirely  of  steel, 
its  walls  being  of  steel  plates  from  6  to  18  in.  thick.    Explain  how  it 
can  float. 

5.  Will  the  water  line  of  a  boat  rise  or  fall  as  it  passes  from-'fresh 
into  salt  water? 

6.  Tf  a  150-lb.  man  can  just  float,  what  is  his  volume? 

7.  A  hollow  steel  body  weighing  1  kg.  just  floats.    What  is  its 
volume  ? 

8.  What  is  the  volume  of  a  whale  which  weighs  30  tons? 

9.  If  each  boat  of  a  pontoon  bridge  is  100  ft.  long  and  75  ft.  wide 
at  the  water  line,  how  much  will  it  sink  when  a  locomotive  weighing 
100  tons  passes  over  it? 

10.  A  block  of  wood  10  in.  high  sinks  6  in.  in  water.  Find  the  density 
of  the  wood.  , 

11.  If  this  block  sank  7  in.  in  oil,  what  would  be  the  density  of 
the  oil? 

12.  To  what  depth  would  it  sink  in  turpentine  of  density  .87? 

13.  A  graduated  glass  cylinder  contains  190  cc.  of  water.    An  egg 
weighing  40  g.  is  dropped  into  the  glass ;  it  sinks  to  the  bottom  and 
raises  the  water  to  the  225-cc.  mark.    Find  the  density  of  the  egg. 

14.  A  cube  of  iron  10  cm.  on  a  side  weighs  7500  g.    What  will  it 
weigh  in  alcohol  of  density  .82? 

*  Laboratory  experiments  on  the  determination  of  the  densities  of  solids  and 
liquids  should  follow  or  accompany  the  discussion  of  this  chapter.  See,  for  example, 
Experiments  7  and  8  of  the  authors'  manual. 


CHAPTER  TTI 


PRESSURE  IN  AIR 
BAROMETRIC  PHENOMENA 

35.  The  weight  of  air.  To  ordinary  observation  air  is 
scarcely  perceptible.  It  appears  to  have  no  weight  and  to  offer 
no  resistance  to  bodies  passing  through  it.  But  if  a  bulb  be 
balanced  as  in  Fig.  23,  then  removed  and  filled  with  air  under 
pressure  by  a  few  strokes  of  a  bicycle  pump,  it  will  be  found, 
when  placed  on  the  balance  again, 
to  be  heavier  than  it  was  before. 
On  the  other  hand,  if  the  bulb  be 
connected  with  an  air  pump  and 
exhausted,  it  will  be  found  to  have 
lost  weight.  Evidently,  then,  air 
can  be  put  into  and  taken  out  of  a 
vessel,  weighed,  and  handled,  just 
like  a  liquid  or  a  solid. 

We  are  accustomed  to  say  that 
bodies  are  "  as  light  as  air,"  yet 
careful  measurement  shows  that  it 
takes  but  12  cubic  feet  of  air  to  weigh  a  pound,  so  that  a 
single  large  room  contains  more  air  than  an  ordinary  man 
can  lift.  Thus  the  air  in  a  room  60  feet  by  30  feet  by  15  feet 
weighs  more  than  a  ton.  The  exact  weight  of  air  at  the 
freezing  temperature  and  under  normal  atmospheric  condi- 
tions is  .001293  gram  per  cubic  centimeter,  that  is,  1.293  grams 
per  liter.  A  given  volume  of  air  therefore  weighs  -^  as  much 
as  an  equal  volume  of  water. 

26 


FIG.  23.    Proof  that  air  has 
weight 


BAKOMETBIC  PHENOMENA  27 

36.  Proof  that  air  exerts  pressure.    Since  air  has  weight, 
it  is  to  be  inferred  that  air,  like  a  liquid,  exerts  force  against 
any    surface    immersed    in    it.     The    following   experiments 
prove  this. 

Let  a  rubber  membrane  be  stretched  over  a  glass  vessel,  as  in  Fig.  24. 
As  the  air  is  exhausted  from  beneath  the  membrane  the  latter  will  be 
observed  to  be  more  and  more  depressed  until 
it  will  finally  burst  under  the  pressure  of  the 
air  above. 

Again,  let  a  tin  can  be  partly  filled  with 
water  and  the  water  boiled.  The  air  will  be 
expelled  by  the  escaping  steam.  While  the 
boiling  is  still  going  on,  let  the'  can  be  tightly 
corked,  then  placed  in  a  sink  or  tray  and  cold 
water  poured  over  it.  The  steam  will  be  con-  FKJ.  24.  Rubber  mem- 
densed  and  the  weight  of  the  air  outside  will  brane  stretched  by  weight 
crush  the  can.  of  air 

37.  Cause  of  the  rise  of  liquids/ in  exhausted  tubes.    If  the 
lower  end  of  a  long  tube  be  dipped  into  water  and  the  air 
exhausted  from  the  upper  end,  water  will,  rise  in  the  tube.   We 
prove  the  truth  of  this  statement  every  time  we  draw  lemonade 
through  a  straw.    The  old  Greeks  and  Romans  explained  such 
phenomena  by  saying  that  "  nature  abhors  a  vacuum,"  and 
this  explanation  was  still  in  vogue  in  Galileo's  time.    But  in 
1640  the  Duke  of  Tuscany 'had  a  deep  well  dug  near  Florence, 
and  found  to  his  surprise  that  no  water  pump  which  could 
be  obtained  would  raise  the  water  higher  than  about  32  feet 
above  the  level  in  the  well.    When  he  applied  to  the  aged 
Galileo  fdr  an  explanation,  the  latter  replied  that  evidently 
"  nature's  horror  of  a  vacuum  did  not  extend  beyond  32  feet." 
It  is  quite  likely  that  Galileo  suspected  that  the  pressure  of 
the  air  was  responsible  for  the  phenomenon,  for  he  had  him- 
self proved  before  that  air  had  weight,  and,  furthermore,  he 
at  once  devised  another  experiment  to  test,  as  he  said,  the 
"power  of  a  vacuum."    He  died  in  1642  before  the  experiment 


28 


PKESSUKE  IN  AIR 


(1) 


was  performed,  but  suggested  to  his  pupil,  Torricelli,  that  he 
continue  the  investigation. 

38.  Torricelli's  experiment.    Torricelli  argued  that  if  water 
would  rise  32  feet,  then  mercury,  which  is  about  13  times  as 
heavy  as  water,  ought  to  rise  but  y^-  as  high.    To  test  this 

inference  he  performed  in  1643  the 
following  famous  experiment :  s 

Let  a  tube  about  4  feet  long,  which  is 
sealed  at  one  end,  be  completely  filled  with 
mercury,  as  in  Fig.  25,  (1),  then  closed  with 
the  thumb  and  inverted,  and  the  bottom 
then  immersed  in  a  dish  of  mercury,  as  in 
Fig.  25,  (2).  When  the  thumb  is  removed 
from  the  bottom  of  the  tube,  the  mercury 
will  fall  away  from  the  upper  end  of  the 
tube  in  spite  of  the  fact  that  in  so  doing 
it  will  leave  a  vacuum  above  it,  and  its 
upper  surface  will,  in  fact,  stand  about 
-i.  of  32  feet,  that 
is,  between  29  and 
30  inches  above  the 
mercury  in  the  dish. 

Torricelli   con- 
cluded  from  this 

experiment  that  the  rise  of  liquids  in  ex- 
hausted tubes  is  due  to  an  outside  pressure 
exerted  by  the  atmosphere  on  the  surface 
of  the  liquid,  and  not  to  any  mysterious 
sucking  power  created  bv  the  vacuum. 

J  FIG.  26.f  Barometer 

39.  Further  decisive  tests.   Anunanswer-    fails  when  air  pres- 
able  argument  in  favor  of  this  conclusion    sure  on  the  mercury 
will  be  furnished  if  the  mercury  in  the  tube      surface  is  reduced 
falls  as  soon  as  the  air  is  removed  from  above  the  surface  of 
the  mercury  in  the  dish. 

To  test  this  point,  let  the  dish  and  tube  be  placed  on  the  table  of  an 
air  pump,  as  in  Fig.  26,  the  tube  passing  through  a  tightly  fitting  rubber 


FIG.  25.   Torricelli's  experi- 
ment 


BAROMETRIC  PHENOMENA 


29 


stopper  A  in  the  bell  jar.  As  soon  as  the  pump  is  started,  the  mercury 
in  the  tube  will,  in  fact,  be  seen  to  fall.  As  the  pumping  is  continued 
it  will  fall  nearer  and  nearer  to  the  level  in  the  dish,  although  it  will 
not  usually  reach  it  for  the  reason  that  an  ordinary  vacuum  pump  is 
not  capable  of  producing  as  good  a  vacuum  as  that  which  exists  in  the 
top  of  the  tube.  As  the  air  is  allowed  to  return  to  the  bell  jar  the 
mercury  will  rise  in  the  tube  to  its  former  level. 


40.  Amount  of  the  atmospheric  pressure.  Torricelli's  ex- 
periment shows  exactly  how  great  the  atmospheric  pressure 
is,  since  this  pressure  is  able  to  balance  a  column  of  mercury 
of  definite  length.  In  accordance  with 
Pascal's  law  the  downward  pressure  ex- 
erted by  the  atmosphere  on  the  surface 
of  the  mercury  in  the  dish  (Fig,  27)  is 
transmitted  as  an  exactly  equal  upward 
pressure  on  the  layer  of  mercury  inside 
the  tube  at  the  same  level  as  the  mercury 
outside.  But  the  downward  pressure  at 
this  point  within  the  tube  is  equal  to  Ttc?, 
where  d  is  the  density  of  mercury  and  li 
is  the  depth  below  the  surface  b.  Since 
the  average  height  of  this  column  at  sea> 
level  is  found  to  be  76  centimeters,  and 
since  the  density  of  mercury  is  13.6,  the 


FIG.  27.    Air   column 
to  top  of  atmosphere 


downward  pressure  inside  the  tube  at  a  is    balance*  the  mfcury 

column  ab 

equal  to  76  times  13.6,  or  1033.6  grams  per 
square  centimeter.    Hence  the  atmospheric  pressure  acting  on 
the  surface  of  the  mercury  in  the  dish  is  1033.6  grams,  or 
roughly   1  kilogram,  per   square  centimeter.     This  amounts 
to  about  15  pounds  per  square  inch. 

41.  PascaPs  experiment.  Pascal  thought  of  another  way  of 
testing  whether  or  not  it  were  indeed  the  weight  of  the  outside 
air  which  sustains  the  column  of  mercury  in  an  exhausted  tube. 
He  reasoned  that,  since  the  pressure  in  a  liquid  diminishes  on 


30 


PRESSURE 


AIR 


ascending  toward  the  surface,  atmospheric  pressure  ought  also 

to  diminish  on  passing  from  sea  level  to  a  mountain  top.  As  no 

mountain  existed  near  Paris,  he  carried  Torricelli's  apparatus 

to  the  top  of  a  high  tower  and  found,  indeed,  a         / — ^ 

slight  fall  in  the  height  of  the  column  of  mercury. 

He  then  wrote  to  his  brother-in-law,  Perrier,  who 

lived  near  Puy  de  Dome,  a  mountain  in  the  south 

of  France,  and  asked  him  to  try  the  experiment 

on  a  larger  scale.   Perrier  wrote  back  that  he  was 

"  ravished  with  admiration   and   astonishment " 

when  he  found  that  on  ascending  1000  meters 

the  mercury  sank  about  8  centimeters  in  the  tube. 

This  was  in  1648,  five  years  after  Torricelli's 

discovery. 

At  the  present  day  geological  parties  actually 
ascertain  differences  in  altitude  by  observing  the 
change  in  the  barometric  pressure  as  they  ascend 
or  descend.  A  fall  of  1  millimeter  in  the  baro- 
metric height  corresponds  to  an  ascent  of  about 
12  meters. 

42.  The  barometer.  The  modern  barometer 
(Fig.  28)  is  essentially  nothing  more  nor  less 
than  Torricelli's  tube.  Taking  a  barometer  read- 
ing consists  simply  in  accurately  measuring  the 
height  of  the  mercury  column.  This  height  varies 
from  73  to  76.5  centimeters  in  localities  which 
are  not  far  above  sea  level,  the  reason  being  that 
disturbances  in  the  atmosphere  affect  the  pres- 
sure at  the  earth's  surface  in  the  same  way  in 
which  eddies  and  high  waves  in  a  tank  of  water  would  affect 
the  liquid  pressure  at  the  bottom  of  the  tank. 

The  barometer  does  not  directly  foretell  the  weather,  but 
it  has  been  found  that  a  low  or  rapidly  falling  pressure  is 
usually  accompanied,  or  soon  followed,  by  stormy  conditions. 


FIG.  28.   The 
barometer 


BAROMETRIC  PHENOMENA 


31 


Hence    the    barometer,   although   not   an  infallible  weather 
prophet,  is  nevertheless    of  considerable  assistance  in  fore- 
casting weather  conditions  some  hours  ahead.    Further,  by 
comparing  at  a  central  station  the  telegraphic 
reports  of  barometer  readings  made  every  few 
hours  at  stations  all  over  the  country,  it  is 
possible  to  determine  in  what  direction  the 
atmospheric   eddies  which   cause   barometer 
changes  and  stormy  conditions  are  traveling, 
and  hence  to  "  forecast "  the  weather  even  a 
day  or  two  in  advance. 


FIG.  29.  Effect  of 
inclining     a     ba- 
rometer tube 


43.  The  first  barometers.    Torricelli  actually  con- 
structed a  barometer  not  essentially  different  from 
that  shown  in  Fig.  28,  and  used  it  for  observing 
changes  in  the  atmospheric  pressure;  but  perhaps 
the  most  interesting  of  the  early  barometers  was  that 
set  up  about  1650  by  the  famous  old  German  physi- 
cist Otto  von  Guericke  of  Magdeburg  (1602-1686).    He  used  for  his  ba- 
rometer a  water  column  the  top  of  which  passed  through  the  roof  of  his 
house.    A  wooden  image  which  floated  on  the  upper  surface  of  the  water 
appeared  above  the  housetop  in  fair  weather  but  retired  from  sight  in 
foul,  a  circumstance  which  led  his  neighbors 

to  charge  him  with  being  in  league  with  Satan. 

44.  Effect  of  inclining  a  barometer. 

If  a  barometer  tube  is  inclined  in  the 
manner  shown  in  Fig.  29,  the  top  of 
the  mercury  will  be  found  to  remain 
always  in  the  same  horizontal  plane. 
Explain,  remembering  that  pressure 
equals  height  times  density  (Fig.  5). 


FIG.  30.  An  aneroid 
barometer 


45.  The  aneroid  barometer.  Since  the  mer- 
curial barometer  is  somewhat  long  and  in- 
convenient to  carry,  geological  and  surveying  parties  commonly  use  an 
instrument  called  the  aneroid  barometer  (Fig.  30).  It  consists  essentially 
of  an  air-tight  cylindrical  box  D,  the  top  of  which  is  a  metallic  diaphragm 


32  PRESSURE  IK  AIR 

which  bends  slightly  under  the  influence  of  change  in  the  atmospheric 
pressure.  This  motion  of  the  top  of  the  box  is  multiplied  by  a  delicate 
system  of  levers  and  communicated  to  a  hand  B,  which  moves  over  a 
dial  whose  readings  are  made  to  correspond  to  the  readings  of  a  mercury 
barometer.  These  instruments  are  made  so  sensitive  as  to  indicate  a 
change  in  pressure  when  they  are  moved  no  farther  than  from  a  table 
to  the  floor. 

QUESTIONS  AND  PROBLEMS 

1.  What  is  your  explanation  of  why  "nature  abhors  a  vacuum  "? 

2.  Find  the  weight  of  the  air  contained  in  a  room  18  x  12  x  4  .-5  m. 

3.  If  a  barometer  were  sunk  in  water  so  that  the  lower  mercury  surface 
stood  1  m.  below  the  surface  of  the  water,  what  would  be  the  reading  of 
the  barometer,  the  barometric  height  at  the  surface  being  75.42  cm.? 

4.  If  a  circular  piece  of  wet  leather,  having  a  string  attached  to  the 
middle,  is  pressed  down  011  a  flat  smooth  stone,  as  in  Fig.  31,  the  latter 
may  often  be  lifted  by  pulling  on  the  string.    Is  it  pulled  up  or  pushed 
up  ?   Explain. 


FIG.  31  FIG.  32  FIG.  33  FIG.  34.    Magdeburg 

hemispheres 

5.  Why  does  not  the  ink  run  out  of  a  pneumatic  inkstand  like  that 
shown  in  Fig.  32  ? 

6.  If  the  variation  of  the  height  of  a  mercury  barometer  is  2  in., 
how  far  did  the  image   rise   and  fall  in  Otto  von  Guericke's  water 
barometer?    (See  §43.) 

7.  If  a  tumbler  is  filled  with  water,  and  a  piece  of  writing  paper  is 
placed  over  the  top,  it  may  be  inverted,  as  in  Fig.  33,  without  spilling 
the  water.    Explain.   What  is  the  function  of  the  paper  ? 

8.  Magdeburg  hemispheres  (Fig.  34)  are  so  called  because  they  were 
invented  by  Otto  von  Guericke,  who  was  mayor  of  Magdeburg.    When 
the  lips  of  the  hemispheres  are  placed  in  contact  and  the  air  exhausted 
from  between  them,  it  is  found  very  difficult  to  pull  them  apart.   Why  ? 

9.  Von  Guericke's  original  hemispheres  are  still  preserved  in  the 
museum  at  Berlin.    Their  interior  diameter  is  22  in.    On  the  cover  of 


OTTO  VON  GUERICKE  (1602-1686) 

German  physicist,  astronomer,  and  man  of  affairs ;  mayor  of  Mag- 
deburg; invented  the  air  pump  in  1650,  and  performed  many 
new  experiments  with  liquids  and  gases ;  discovered  electrostatic 
repulsion ;  constructed  the  famous  Magdeburg  hemispheres  which 
four  teams  of  horses  could  not  pull  apart  (see  p.  32) 


COMPRESSIBILITY  OF  AIR  33 

the  book  which  describes  his  experiments  is  a  picture  which  represents 
4  teams  of  horses  on  each  side  of  the  hemispheres  trying  to  separate 
them.  The  experiment  was  actually  performed  in  this  way  before  the 
German  emperor  Ferdinand  III.  If  the  air  was  all  removed  from  the 
interior  of  the  hemispheres,  what  force  in  pounds  was  in  fact  required  to 
pull  them  apart  ?  (Find  the  atmospheric  force  on  a  circle  of  11  in.  radius.) 

COMPRESSIBILITY  AND  EXPANSIBILITY  OF  AIR 

46.  Incompressibility  of  liquids.    Thus  far  we  have  found 
very  striking  resemblances  between  the  conditions  which  exist 
at  the  bottom  of  a  body  of  liquid  and  those  which  exist  at  the 
bottom  of  the  great  ocean  of  air  in  which  we  live.    We  now 
come  to  a  most  important  difference.    It  is  well  known  that 
if  2  liters  of  water  be  poured  into  a  tall  cylindrical  vessel,  the 
water  will  stand  exactly  twice  as  high  as  if  the  vessel  contain 
but  1  liter;  or  if  10  liters  be  poured  in,  the  water  will  stand 
10  times  as  high  as  if  there  be  but  1  liter.    This  means  that 
the  lowest  liter  in  the  vessel  is  not  measurably  compressed 
by  the  weight  of  the  water  above  it. 

It  has  been  found  by  carefully  devised  experiments  that 
compressing  weights  enormously  greater  than  these  may  be 
used  without  producing  a  marked  effect ;  namely,  when  a 
cubic  centimeter  of  water  is  subjected  to  the  stupendous  pres- 
sure of  3,000,000  grams,  its  volume  is  reduced  to  but  .90  cubic 
centimeter.  Hence  we  say  that  water,  and  liquids  generally, 
are  practically  incompressible.  Had  it  not  been  for  this  fact 
we  should  not  have  been  justified  in  taking  the  pressure  at 
any  depth  below  the  surface  of  the  sea  as  the  simple  product 
of  the  depth  by  the  density  at  the  surface. 

47.  Compressibility  of  air.    When  we  study  the  effects  of 
pressure  on  air  we  find  a  wholly  different  behavior  from  that 
described  above  for  water.    It  is  very  easy  to  compress  a  body 
of  air  to  one  half,  one  fifth,  or  one  tenth  of  its  normal  volume, 
as  we  prove  every  time  we  inflate  a  pneumatic  tire  or  cushion  of 
any  sort.    Further,  the  expansibility  of  air,  that  is,  its  tendency 


34  PEESSUEE  IN  AIE 

to  spring  back  to  a  larger  volume  as  soon  as  the  pressure 
is  relieved,  is  proved  every  time  a  tennis  ball  or  a  football 
bounds,  or  the  cork  is  driven  from  a  popgun. 

But  it  is  not  only  air  which  has  been  crowded  into  a  pneu- 
matic cushion  by  some  sort  of  a  pressure  pump  which  is  in 
this  state  of  readiness  to  expand  as  soon  as  the  pressure  is 
diminished ;  the  ordinary  air  of  the  room  will  expand  in  the 
same  way  if  the  pressure  to  which  it  is  subjected  is  relieved. 

Thus,  let  a  liter  beaker  with  a  sheet  of  rubber  dam  tied  tightly 
over  the  top  be  placed  under  the  receiver  of  an  air  pump.  As  soon 
as  the  pump  is  set  into  operation 
the  inside  air  will  expand  with 
sufficient  force  to  burst  the  rub- 
ber or  greatly  to  distend  it,  as 
shown  in  Fig.  35. 

Again,  let  two  bottles  be  ar- 
ranged as  in  Fig.  36,  one  being 
stoppered  air-tight,  while  the  FIG.  35  FIG.  36 

other  is  uncorked.    As   soon   as     illustrations  of  the  expansibility  of  air 
'  the    two   are    placed    under    the 

receiver  of  an  air  pump  and  the  air  exhausted,  the  water  in  A  will  pass 
over  into  B.  When  the  air  is  readmitted  to  the  receiver  the  water  will 
flow  back.  Explain. 

48.  Why  hollow  bodies  are  not  crushed  by  atmospheric  pres- 
sure. The  preceding  experiments  show  why  the  walls  of  hol- 
low bodies  are  not  crushed  in  by  the  enormous  forces  which 
the  weight  of  the  atmosphere  exerts  against  them.  For  the 
air  inside  such  bodies  presses  their  walls  out  with  as  much 
force  as  the  outside  air  presses  them  in.  In  the  experiment 
of  §  36  the  inside  air  was  removed  by  the  escaping  steam. 
When  this  steam  was  condensed  by  the  cold  water,  the  inside 
pressure  became  very  small  and  the  outside  pressure  then 
crushed  the  can.  In  the  experiment  shown  in  Fig.  35  it  was 
the  outside  pressure  which  was  removed  by  the  air  pump,  and 
the  pressure  of  the  inside  air  then  burst  the  rubber. 


COMPRESSIBILITY  OF  AIR  35 

49.  Boyle's  law.  The  first  man  to  investigate  the  exact 
relation  between  the  change  in  the  pressure  exerted  by  a  con- 
fined body  of  air  and  its  change  in  volume  was  an  Irishman, 
Robert  Boyle  (1626-1691).  We  shall  repeat 
a  modified  form  of  his  experiment  much  more 
carefully  in  the  laboratory ;  but  the  follow- 
ing will  illustrate  the  method  by  which  he 
discovered  one  of  the  most  important  laws 
of  physics. 

Let  mercury  be  poured  into  a  bent  glass  tube  until 
it  stands  at  the  same  level  in  the  closed  arm  AC  as 
in  the  open  arm  BD  (Fig.  37).    There  is  now  con- 
fined in  AC  a  certain  volume  of  air  under  the  pres- 
sure of  one  atmosphere.    Call  this  pressure  Pr    Let 
the  length  A  C  be  measured  and  called  Vr    Then  let 
mercury  be  poured  into  the  long  arm  until  the  level     T?   .    QT     M  tl    d 
in  this  arm  is  as  many  centimeters  above  the  level  in     ()£  demonstrating 
the  short  arm  as  there  are  centimeters  in  the  barom-          Boyle's  law 
eter  height.    The  confined  air  is  now  under  a  pressure 
of  two  atmospheres.    Call  it  P2.    Let  the  new  volume  A^C(=  Tr2)  be 
measured.    It  will  be  found  to  be  just  half  its  former  value. 

Hence  we  learn  that  doubling  the  pressure  exerted  upon  a 
body  of  air  halves  its  volume.  If  we  had  tripled  the  pressure, 
we  should  have  found  the  volume  reduced  to  one  third  its 
initial  value,  etc.  That  is,  the  pressure  which  a  given  quantity 
of  air  at  constant  temperature  exerts  against  the  walls  of  the  con- 
taining vessel  is  inversely  proportional  to  the  volume  occupied. 
This  is  (algebraically  stated  thus : 

P       V  ' 


--B 


This  is  Boyle's  law.  It  may  also  be  stated  in  slightly  differ- 
ent form.  Doubling,  tripling,  or  quadrupling  the  pressure 
must  double,  triple,  or  quadruple  the  density,  siiicethe  volume 
is  made  only  one  half,  one  third,  or  one  fourth  as  much,  while 

t 


36  PRESSURE  IN  AIR 

the  mass  remains  unchanged.    Hence  the  pressure  which  air 
exerts  is  directly  proportional  to  its  density,  or,  algebraically, 

P       D   * 
— *  =  — '• 
P        7) 

^1  JJ-2 

50.  Extent  and  character  of  the  earth's  atmosphere.  From 
the  facts  of  compressibility  and  expansibility  of  air  we  may 
know  that  the  air,  unlike  the  sea,  must  become  less  and  less 
dense  as  we  ascend  from  the  bottom  toward  the  top.  Thus 
at  the  top  of  Mont  Blanc,  where  the  barometer  height  is  but 
38  centimeters,  or  -one  half  of  its  value  at  sea  level,  the  den- 
sity also  must,  by  Boyle's  law,  be  just  one  half  as  much  as  at 
sea  level* 

No  one  has  ever  ascended  higher  than  7  miles,  which  was 
approximately  the  height  attained  in  1862  by  the  two  daring 
English  aeronauts,  Glasier  and  Coxwell.  At  this  altitude  the 
barometric  height  is  but  about  7  inches,  and  the  temperature 
about  —  60°  F.  Both  aeronauts  lost  the  use  of  their  limbs 
and  Mr.  Glasier  became  unconscious.  Mr.  Coxwell  barely 
succeeded  in  grasping  with  his  teeth  the  rope  which  opened  a 
valve  and  caused  the  balloon  to  descend.  Again,  on  July  31, 
1901,  the  French  aeronaut  M.  Berson  rose  without  injury  to 
a  height  of  about  7  miles  (35,420  feet),  his  success  being  due 
to  the  artificial  inhalation  of  oxygen. 

By  sending  up  self -registering  thermometers  and  barometers 
in  balloons  which  burst  at  great  altitudes,  the  instruments 
being  protected  by  parachutes  from  the  dangers  of  rapid 
fail,  the  atmosphere  has  been  explored  to  a  height  of  30,500 
meters  (18.95  miles),  this  being  the  height  attained  on  Sep- 
tember 1,  1910,  by  a  little  balloon  holding  6.2  cubic  meters  of 
hydrogen  at  the  surface,  which  was  sent  up  at  Huron,  South 
Dakota,  by  William  R.  Blair  of  the  United  States  Government 

*  A  laboratory  experiment  on  Boyle's  law  should  follow  this  discussion.  See, 
for  example,  Experiment  9  of  the  authors'  manual. 


COMPRESSIBILITY  OF  AIR 


37 


Observatory,  Mount  Weather,  Virginia.  These  extreme  heights 
are  calculated  from  the  indications  of  the  self-registering 
barometers. 

At  a  height  of  35  miles  the  density  of  the  atmosphere  is 


estimated  to  be  but 


30000 


of  its  value  at  sea  level.    By  calcu- 


lating how  far  below  the  horizon  the  sun  must  be  when  the 


35-- 


10-- 


--7* 


FlG. 


30  1 

Extent  and  character  of  atmosphere 


densities 


last  traces  of  color  disappear  from  the  sky,  we  find  that  at  a 
height  as  great  as  45  miles  there  must  be  air  enough  to  reflect 
some  light.  How  far  beyond  this  an  extremely  rarified  atmos- 
phere may  extend,  no  one  knows.  It  has  been  estimated  at 
all  the  way  from  100  to  500  miles.  These  estimates  are  based 
on  observations  of  the  height  at  which  meteors  first  become 
visible,  on  the  height  of  the  aurora  borealis,  and  on  the  dark- 
ening of  the  surface  of  the  moon  just  before  it  is  eclipsed  by 
the  shadow  of  the  solid  earth. 


38  PEESSUEE  IN  AIE 

51.  Height  of  the  "  homogeneous  atmosphere."   Although, 
then,  we  cannot  tell  to  what  height  the  atmosphere  extends, 
we  do  know  with  certainty  that  the  weight  of  a  column  of  air 
1  square  centimeter  in  cross  section  and  reaching  from  the 
earth's  surface  to  the  extreme  limits  of  the  atmosphere  will 
just  balance  a  column  of  mercury  76  centimeters  high,  for 
this  was  shown  by  Torricelli's  experiment.    Since  1  cubic  cen- 
timeter of  air  at  the  earth's  surface  Aveighs  about  1.2  milli- 
grams, that  is,  since  the  density  of  air  is  about  .0012,  or  — |-^ 
that  of  water,   and  since  mercury   is   about   13.6    times   as 
heavy  as  water,  it  follows  that  if  the  air  had  the  same  density 
at  all  altitudes  which  it  has  at  the  earth's  surface,  its  height 
would  be  76  x  13.6  x  800  centimeters,  that  is,  8.2  kilometers, 
or  about  5  miles.    The  tops  of  the  Himalayas  would  there- 
fore rise  above  it.    This  height  of  5  miles,  which  is  the  height 
to  which  the  air  would  extend  if  it,  like  the  ocean,  had  the 
same  density  throughout,  is  called  the  height  of  the  homogeneous 
atmosphere. 

52.  Density  of  air  below  sea  level.  The  same  cause  which 
makes  air  diminish  rapidly  in  density  as  we  ascend  above  sea 
level  must  produce  a  rapid  increase  in  its  density  as  we  descend 
below  this  level.   It  has  been  calculated  that  if  a  boring  could 
be  made  in  the  earth  35  miles  deep,  the  air  at  the  bottom 
would  be  one  thousand  times  as  dense  as  at  the  earth's  surface. 
Therefore  wood  and  even  water  would  float  in  it. 


QUESTIONS  AND  PROBLEMS 

1.  Why  is  the  reading  of  a  barometer  incorrect  if  the  barometer 
tube  is  not  strictly  vertical  ? 

2.  Under  ordinary  conditions  a  gram  of  air  occupies  about  800  cc. 
Find  what  volume  a  gram  will  occupy  at  the  top  of  Mont  Blanc  (alti- 
tude 15,810  ft.),  where  the  barometer  indicates  that  the  pressure  is  only 
about  one  half  what  it  is  at  sea  level. 

3.  The  mean  density  of  the  air  at  sea  level  is  about  .0012.    What 
is  its  density  at  the  top  of  Mont  Blanc  ?    What  fractional  part  of  the 


PNEUMATIC  APPLIANCES  39 

earth's  atmosphere  has  one  left  beneath  him  when  he  ascends  to  the 
top  of  this  mountain? 

4.  If  Glasier  and  Coxwell  rose  in  their  balloon  until  the  barometric 
height  was  only  18  cm.,  how  many  inhalations  were  they  obliged  to 
make  in  order  to  obtain  the  same  amount  of  air  which  they 

could  obtain  at  the  surface  in  one  inhalation? 

5.  With  the  aid  of  the  experiment  in  which  the  rubber  dam 
was  burst  under  the  exhausted  receiver  of  an  air  pump,  explain 
why  high  mountain  climbing  often  causes  pain  and  bleeding  in 
the  ears  and  nose.  Why  does  deep  diving  produce  similar  effects  ? 

6.  Blow  as  hard  as  possible  into  the  tube  of  the  bottle 
shown  in  Fig.  39.    Then  withdraw  the  mouth  and  explain  all      '     ' 
of  the  effects  observed. 

7.  If  a  bottle  or  cylinder  is  filled  with  water  and  inverted  in  a  dish 
of  water,  with  its  mouth  beneath  the  surface  (see  Fig.  40),  the  water 
will  not  run  out.    Why  ? 

8.  If  a  bent  rubber  tube  is  inserted  beneath  the  cylinder  arid  air 
blown  in  at  o  (Fig.  40),  it  will  rise  to  the  top  and  displace  the  water. 
This  is  the  method  regularly  used  in  collecting  gases.    Explain  what 
forces  the  gas  up  into  it,  and  what  causes  the  water  to 

descend  in  the  tube  as  the  gas  rises. 

9.  Why  must  the  bung  be  removed  from   a  cider 
barrel  in  order  to  secure  a  proper  flow  from  the  faucet? 

10.  When  a  bottle  full  of  water  is  inverted,  the  water 
will  gurgle  out  instead  of  issuing  in  a  steady  stream. 
Why? 

11.  There  is  a  pressure  of  80  cm.  of  mercury  on  1000  cc. 

of  gas.    What  pressure  must  be  applied  to  reduce  the          -pIG  40 
volume  to  600  cc.  if  the  temperature  is  kept  constant  ? 

12.  What  sort  of  a  change  in  volume  do  the  bubbles  of  air  which 
escape  from  a  diver's  suit  experience  as  they  ascend  to  the  surface  ? 

PNEUMATIC  APPLIANCES 

53.  The  siphon.  Let  a  rubber  or  glass  tube  be  filled  with  water  and 
then  placed  in  the  position  shown  in  Fig.  41.  Water  will  be  found  to 
flow  through  the  tube  from  vessel  A  into  vessel  B.  If,  then,  B  be  raised 
until  the  water  in  it  is  at  a  higher  level  than  that  in  A,  the  direction 
of  flow  will  be  reversed.  This  instrument,  which  is  called  the  siphon,  is 
very  useful  for  removing  liquids  from  vessels  which  cannot  be  over- 
turned, or  for  drawing  off  the  upper  layers  of  a  liquid  without  disturbing 
the  lower  layers. 


40 


PRESSURE  IN  AIK 


The  explanation  of  the  siphon's  action  is  readily  seen  from 
Fig.  41.  Since  the  tube  acb  is  full  of  water,  water  must  evi- 
dently flow  through  it  if  the  force  which  pushes  it  one  way 
is  greater  than  that  which  pushes  it  the  other  way.  Now  the 
upward  pressure  at  a  is  equal  to  atmos- 
pheric pressure  minus  the  downward  pres- 
sure due  to  the  water  column  ad\  while 
the  upward  pressure  at  b  is  the  atmospheric 
pressure  minus  the  downward  pressure  due 
to  the  water  column  be.  Hence  the  pressure 
at  a  exceeds  the  pressure  at  b  by  the  pres- 
sure due  to  the  water  column  fb.  The  Fm  41  The  sipholl 
siphon  will  evidently  cease  to  act  when  the 
water  is  at  the  same  level  in  the  two  vessels,  since  then  fb  =  0, 
and  the  forces  acting  at  the  two  ends  of  the  tube  are  therefore 
equal  and  opposite.  It  will  also  cease  to  act  when  the  bend  c 
is  more  than  34  feet  above  the  surface  of 
the  water  in  A,  since  then  a  vacuum  will 
form  at  the  top,  atmospheric  pressure  being 
unable  to  raise  water  to  a  height  greater 
than  this  in  either  tube. 

Would  a  siphon  flow  in  a  vacuum  ? 


FIG.  42.   Intermittent 
siphon 


54.  The   intermittent   siphon.    Fig.  42    repre- 
sents an   intermittent  siphon.    If  the  vessel  is 

at  first  empty,  to  what  level  must  it  be  filled  before  the  water  will 
flow  out  at  o  ?  To  what  level  will  the  water  then  fall  before  the  flow 
will  cease? 

55.  The  air  pump.  The  air  pump  was  invented  in  1650  by 
Otto  von  Guericke,  mayor  of  Magdeburg,  Germany,  who  de- 
serves the  greater  credit,  since  he  was  apparently  wholly  with- 
out knowledge  of  the  discoveries  which  Galileo,  Torricelli, 
and  Pascal  had  made  a  few  years  earlier  regarding  the  char- 
acter of  the  earth's  atmosphere.    A  simple  form  of  such  a 
pump  is  shown  in  Fig.  43.    When  the  piston  is  raised  the  air 


PNEUMATIC  APPLIANCES 


41 


from  the  receiver  R  expands  into  the  cylinder  B  through  the 
valve  A.  When  the  piston  descends  it  compresses  this  air, 
and  thus  closes  the  valve  A  and  opens  the  exhaust  valve  C. 
Thus  with  each  double  stroke  a  certain  fraction  of  the  air 
in  the  receiver  is  transferred  from  R 
through  the  cylinder  to  the  outside. 

In  many  pumps  the  valve   C  is  in 
the  piston  itself. 

56.  The  compression  pump.   A  com- 
pression pump  is  nothing  but  an  ex- 
haust pump  with  the  valves  reversed, 

so  that  A  closes  and  C  opens  on  the    FIG  43    A  simple  air  pump 
upstroke,  and  A  opens  and  C  clcses 

on  the  downstroke.  In  its  cheaper  forms,  for  example,  the 
common  bicycle  pump,  the  valve  C  is  often  replaced  by  a  very 
simple  device  called  a  cup  valve.  This  valve  consists  of  a 
disk  of  leather  a  little  larger  than  the  barrel  of  the  pump, 
attached  to  a  loosely  fitting  metal  piston.  When  the  piston 
is  raised  the  air  passes  in  around  the  leather,  but  when  it  is 
lowered  the  leather  is  crowded  closely  against 
the  walls,  so  that  there  is  no  escape  for  the 
air  (Fig.  44). 

Compressed  air  finds  so  many  applications 
in  such  machines  as  air  drills  (used  in  min- 
ing),  air  brakes,  air  motors,   etc.,  that  the    FlG-  44-  The  CUP 
compression  pump  must  be  looked  upon  as  of 
much  greater  importance  industrially  than  the  exhaust  pump. 

57.  The  lift  pump.    The  common  water  pump,  shown  in 
Fig.  45,  has  been  in  use  at  least  since  the  time  of  Aristotle 
(fourth  century  B.C.).    It  will  be  seen  from  the  figure  that  it 
is  nothing  more  nor  less  than  a  simplified  form  of  air  pump. 
In  fact,  in  the  earlier  strokes  we  are  simply  exhausting  air 
from  the    pipe  below  the  valve   b.    Water  could   never  be 
obtained  at  /S»,  even  with  a  perfect  pump,  if  the  valve  b  were 


42 


PRESSURE  IN  AIR 


not  within  34  feet  of  the  surface  of  the  water  in  W.  Why  ? 
On  account  of  mechanical  imperfections  this  limit  is  usually 
about  28  feet  instead  of  34.  Let  the  student 
analyze,  stroke  by  stroke,  the  operation  of 
pumping  water  from  a  well  with  the  pump 
of  Fig.  45.  Why  will  pouring  in  a  little 
water  at  the  top,  that  is,  "priming,"  often 
assist  greatly  in  starting  such  a  pump  ? 

58.  The  force   pump.    Fig.  46   illustrates 
the  construction  of  the  force  pump,  a  device 

commonly    used    when 

it  is  desired  to  deliver 

water  at  a  point  higher 

than    the    position    at 

which  it  is  convenient 

to  place  the  pump  itself.   Let  the  student 

analyze  the  action  of  the  pump  from  a 

study  of  the  diagram. 

It  will  be  seen  that  the  discharge  from 

such  an  arrangement  as  that  sliown  in 

Fig.  46  must  be  intermittent,  since  no 

water  can  flow  up  the  pipe 

when     the     Piston     P 


FIG.  45.    The  lift 
pump 


FIG.  46.  The  force  pump 

is  ascending.    In   order  to 

make  the  flow  continue  during  the  upstroke,  an 

air  chamber,  such  as  that  shown  in  Fig.  47,  is 

always  inserted  between  the  valve  a  (Fig.  46) 

and  the  discharge  point.    As  the  water  is  forced 

violently  into  this  chamber   by  the  downward 

motion  of  the  piston  it  compresses  the  confined 

air.    It  is,  then,  the  reaction  of  this  compressed 

air  which  is  immediately  responsible  for  the  flow  in  the  dis- 

charge tube,  and  as  this  reaction  is  continuous  the  flow  is 

also  continuous. 


FIG.  47.    The 

air  dome  of  a 

force  pump 


PNEUMATIC  APPLIANCES 


43 


Fig.  48  represents  one  of  the  most  familiar  types  of  force 
pump,  the  double-acting  steam  fire  engine.  Let  the  student 
analyze  the  action  of  the  pump  from  a  study  of  the  diagram. 


Exhaust 


FIG.  48.   The  fire  engine 

59.  The  Cartesian  diver.  Descartes  (1596-1650),  the  great 
French  philosopher,  invented  an  odd  device  which  illustrates  at 
the  same  time  the  principle  of  the  transmission  of  pressure  by 
liquids,  the  principle  of  Archimedes,  and 
the  compressibility  of  gases.  A  hollow 
glass  image  in  human  shape  [Fig.  49,  (1)] 
has  an  opening  in  the  lower  end.  It  is 
partly  filled  with  water  and  partly  with 
air,  so  that  it  will  just  float.  By  press- 
ing on  the  rubber  diaphragm  at  the  top 
of  the  vessel  it  may  be  made  to  sink  or 
rise  at  will.  Explain.  If  the  diver  is  not 
available,  a  small  bottle  or  test  tube  [see 
Fig.  49,  (2)]  may  be  used  instead.  It  works  equally  well,  and 
brings  out  the  principle  even  better.  The  modern  submarine  is 
essentially  nothing  but  a  huge.  Cartesian  diver.  The  volume  of 
the  air  in  its  chambers  is  changed  by  forcing  water  in  or  out. 


FIG.  49.  The  Cartesian 
diver 


44 


PRESSURE  IN  AIR 


FIG.  50.   The  balloon 


60.  The  balloon.  A  reference  to  the  proof  of  Archimedes'  principle 
(§  30,  p.  22)  will  show  that  it  must  apply  as  well  to  gases  as  to  liquids. 
Hence  any  body  immersed  in  air  is  buoyed  up  by 
a  force  which  is  equal  to  the  weight  of  the  displaced 
air.  The  body  will  therefore  rise  if  its  own 
weight  is  less  than  the  weight  of  the  air  which 
it  displaces. 

A  balloon  is  a  large  silk  bag  (Fig.  50)  var- 
nished so  as  to  be  air-tight,  and  filled  either 
with  hydrogen  or  with  common  illuminating  gas. 
The  former  gas  weighs  about  .09  kilogram  per 
cubic  meter  and  common  illuminating  gas  weighs 
about  .75  kilogram  per  cubic  meter.  It  will  be  re- 
membered that  ordinary  air  weighs  about  1.20 
kilograms  per  cubic  meter.  It  will  be  seen,  there- 
fore, that  the  lifting  power  of  hydrogen  per  cubic 
meter,  namely,  1.20  —  .09  =  1.11,  is  more  than 
twice  the  lifting  power  of  illuminating  gas, 
1.20  —  .75  =  .45.  Nevertheless,  on  account  of  the 
comparative  cheapness  of  the  latter  gas  its  use 
is  very  much  more  common. 

Ordinarily  a  balloon  is  not  completely  filled  at  the  start,  for  if 
it'  were,  since  the  outside  pressure  is  continually  diminishing  as  it 
ascends,  the  pressure  of  the  inside  gas  would 
subject  the  bag  to  enormous  strain,  and 
would  surely  burst  it  before  it  reached  any 
considerable  altitude.  But  if  it  is  but  par- 
tially inflated  at  the  start,  it  can  increase  in 
volume  as  it  ascends  by  simply  inflating  to 
a  greater  extent.  Thus  a  balloon  which  as- 
cends until  the  pressure  is  but  7  centimeters 
of  mercury  should  be  only  about  one  fourth 
inflated  when  it  is  at  the  surface  of  the 
.earth. 

The  parachute  seen  hanging  from  the  side 
of  the  balloon  in  Fig.  50  is  a  huge  umbrella- 
like  affair  with  which  the  aeronaut  may  de- 
scend in  safety  to  tjie  earth.  After  opening,  as 
in  Fig.  51,  it  descends  very  slowly  on  account 

of  the  enormous  surface  exposed  to  the  air.   The  hole  in  the  top  allows 
air  to  escape  slowly  and  thus  keeps  the  parachute  upright. 


FIG.  51.    The  parachute 


PNEUMATIC  APPLIANCES 


45 


FIG.  52.    The 
diving  bell 


61.  The  diving  bell.  The  diving  bell  (Fig.  52)  is  a  heavy 
bell-shaped  body  with  rigid  walls,  which  sinks  of  its  own 
weight.  Formerly  the  workmen  who  went  down  in  the  bell 
had  at  their  disposal  only  the  amount  of  air 
confined  within  it,  and  the  water  rose  to  a  cer- 
tain height  within  the  bell  on  account  of  the 
compression  of  the  air.  But  in  modern  practice 
the  air  is  forced  in  from  the  surface  through 
a  connecting  tube  a  (Fig.  53)  by  means  of  a 
force  pump  h.  This  arrangement,  in  addition 
to  furnishing  a  continual  supply  of  fresh  air, 
makes  it  possible  to  force  the  water  down  to  the  level  of 
the  bottom  of  the  bell.  In  practice  a  continual  stream  of 
bubbles  is  kept  flowing  out  from  the  lower  edge  of  the 
bell,  as  shown  in  Fig.  53. 

The  pressure  of  the  air 
within  the  bell  must,  of 
course,  be  the  pressure  ex- 
isting within  the  water  at 
the  depth  of  the  level  of 
the  water  inside  the  bell, 
that  is,  in  Fig.  52  at  the 
depth  A  C.  Thus  at  a  depth 
of  34  feet  the  pressure 
is  2  atmospheres.  Diving 
bells  are  used  for  put- 
ting in  the  foundations  of 
bridge  piers,  doing  subaque- 
ous excavating,  etc.  The  FIG.  53.  Laying  foundations  of  piers 
so-called  caisson,  much  used  with  the  divinS  bel1 

in  bridge  building,  is  simply  a  huge  stationary  diving  bell, 
which  the  workmen  enter  through  compartments  provided 
with  air-tight  doors.  Air  is  pumped  into  it  precisely  as  in 
Fig.  53. 


46 


PRESSURE  IN  AIR 


Fi«.  54.    The  diving 
suit 


62.  The  diving  suit.    For  most  purposes,  except  those  of  heavy  engi- 
neering, the  diving  suit  has  now  replaced  the  diving  bell.    This  suit  is 
made  of  rubber  with  a  metal  helmet.    The  diver 

is  sometimes  connected  with  the  surface  by  a  tube 
(Fig.  54)  through  which  air  is  forced  down  to 
him.  It  passes  out  into  the  water  through  the 
valve  v  in  his  suit.  But  more  commonly  the  diver 
is  entirely  independent  of  the  surface,  carrying 
air  under  a  pressure  of  about  40  atmospheres 
in  a  tank  on  his  back.  This  air  is  allowed  to 
escape  gradually  through  the  suit  and  out  into 
the  water  through  the  valve  v  as  fast  as  the  diver 
needs  it.  When  he  wishes  to  rise  to  the  surface 
he  simply  admits  enough  air  to  his  suit  to  make 
him  float. 

In  all  cases  the  diver  is  subjected  to  the  pres- 
sure existing  at  the  depth  at  which  the  suit  or 
bell  communicates  with  the  outside  water.  Div- 
ers seldom  work  at  depths  greater  than  60  feet, 
and  80  feet  •  is  usually  considered  the  limit  of 
safety.  But  in  building  the  bridge  over  the  Mis- 
sissippi at  St.  Louis,  Missouri,  the  bells  with  their  divers  were  sunk 
to  a  depth  of  110  feet,  while  a  case  is  on  record  of  a  diver  who,  in 
investigating  a  wreck  off  the  coast 
of  South  America,  sank  to  a  depth 
of  201  feet. 

The  diver  experiences  pain  in 
the  ears  and  above  the  eyes  when 
he  is  ascending  or  descending,  but 
not  when  at  rest.  This  is  because 
it  requires  some  time  for  the  air  to 
penetrate  into  the  interior  cavities 
of  the  body  and  establish  equal  pres- 
sure in  both  directions. 

63.  The  air  brake.    Fig.  55  is  a 
diagram  which  shows  the  essential 
features   of    the   Westinghouse    air 
brake.  P  is  an  air  pipe  leading  to  the 

engine,  where  a  compression  pump  maintains  air  in  the  main  cylinder 
under  a  pressure  of  about  70  pounds  to  the  square  inch.  R  is  an  auxiliary 
reservoir  which  is  placed  under  each  car,  and  which  connects  with  P 


FIG.  55.   The  Westinghouse  air 
brake 


PNEUMATIC  APPLIANCES 


47 


through  the  triple  valve  V.  So  long  as  the  pressure  from  the  engine  is 
on  in  P,  the  valve  V  is  open  in  such  a  way  that  there  is  direct  com- 
munication between  P  and  R.  But  as  soon  as  the  pressure  in  P  is 
diminished,  either  by  the  engineer  or  by  the  accidental  breaking  of  the 
hose  coupling  k,  which  connects  P  from  car  to  car,  the  compressed 
air  in  R  operates  the  valve  in  V  so  as  to  shut  off  connection  between 
R  and  P  and  to  open  connection  between  R  and  the  cylinder  C.  The 
piston  //  is  thus  driven  powerfully  to  the  left  and  sets  the  brake  shoes 
against  the  wheels  through  the  operation  of  levers  attached  to  //.  When 
it  is  desired  to  take  off  the  brakes,  pressure  is  again  turned  on  in  P. 
This  operation  opens  V  in  such  a  way  as  to  permit  the  compressed 
air  in  C  to  escape,  and  the  spring  S  then  pulls  back  the  brake  shoes 
from  the  wheels. 

64.  The  bellows.     Fig.  56  shows  the  construction  of  the 
ordinary  blacksmith's  bellows.    When  the  handle  a  rises  and 
the  point  b  in  consequence 

falls,  the  valve  v  opens  and 
air  from  the  outside  enters 
the  lower  compartment  C^ 
When  a  is  pulled  down  and 
b  thus  made  to  ascend,  v  at 
once  closes,  and  as  soon  as 
the  pressure  within  C1  has 
risen  to  the  same  value  as 
that  maintained  in  C2  by  the 

weights  Wj  the  valve  v'  opens  and  air  passes  from  Cl  to  (?2. 
With  this  arrangement  it  will  be  seen  that  the  current  of  air 
which  issues  from  C2  through  the  nozzle  is  continuous  rather 
than  intermittent,  as  it  would  be  if  there  were  but  one  com- 
partment and  one  valve. 

65.  The  gas  meter.    The  gas  meter  is  a  device  which  differs  little  in 
principle  from  the  blacksmith's  bellows.   Gas  from  the  city  supply  enters 
the  meter  through  P  (Fig.  57)  and  passes  through  the  openings  o  and 
ot  into  the  compartments  B  and  Bl  of  the  meter.  Here  its  pressure  forces 
in  the  diaphragms  d  and  clr    The  gas  already  contained  in  A  and  A1is 
therefore  pushed  out  to  the  burners  through  the  openings  o'  and  o[  and 


FIG.  56.    A  ^blacksmith's  bellows 


CHAPTER  IV 
V 

MOLECULAR  MOTIONS 

KINETIC  THEORY  OF  GASES 

66.  Molecular  constitution  of  matter.    In  order  to  account 
for  some  of  the  simplest  facts  in  nature,  —  for  example,  the 
fact  that  two  substances  often  apparently  occupy  the  same 
space  at  the  same  time,  as  when  two  gases  are  crowded  together 
in  the  same  vessel,  or  when  sugar  is  dissolved  in  water,  —  it 
is  now  universally  assumed  that  all  substances  are  composed 
of  very  minute  particles  called  molecules.   Spaces  are  supposed 
to  exist  between  these  molecules,  so  that  when  one  gas  enters 
a  vessel  which  is  already  full  of  another  gas,  the  molecules  of 
the  one  scatter  themselves  about  among  the  molecules  of  the 
other.    Since  molecules  cannot  be  seen  with  the  most  powerful 
microscopes,  it  is  evident  that  they  must  be  very  minute.    The 
number  of  them  contained  in  a  cubic  centimeter  of  air  is  27 
billion  billion  (27  X  1018).    It  would  take  as  many  as  a  thou- 
sand molecules  laid  side  by  side  to  make  a  speck  long  enough 
to  be  seen  with  the  best  microscopes. 

67.  Evidence  for  molecular  motions  in  gases.    Certain  very 
simple  observations  lead  us  to  the  conclusion  that  the  mole- 
cules of  gases,  even  in  a  still  room,  must  be  in  continual  and 
quite  rapid  motion.    Thus,  if  a  little  chlorine,  or  ammonia, 
or  any  gas  of  powerful  odor  is  introduced  into  a  room,  in  a 
very  short  time  it  will  have  become  perceptible  in  all  parts  of 
the  room.    This  shows  clearly  that  enough  of  the  molecules 
of  the  gas  to  effect  the  olfactory  nerves  must  have  found 
their  way  across  the  room. 

50 


KINETIC  THEORY  OF  GASES 


51 


diffusion 
gases 


Again,  chemists  tell  us  that  if  two  globes,  one  containing 
hydrogen  and  the  other  carbon  dioxide  gas,  be  connected  as  in 
Fig.  59  and  the  stopcock  between  them  opened,  after  a  fe\v 
hours  chemical  analysis  will  show  that  each 
of  the  globes  contains  the  two  gases  in  exactly 
the  same  proportions  —  a  result  which  is  at 
first  sight  very  surprising,  since  carbon  diox- 
ide gas  is  about  twenty-two  times  as  heavy 
as  hydrogen.  This  mixing  of  gases  in  appar- 
ent violation  of  the  laws  of  weight  is  called 
diffusion. 

We  see  then  that  such  simple  facts  as  the 
transference  of  odors  and  the  diffusion  of 
gases  furnish  very  convincing  evidence  that  '  FlG-  59-  Ulustrat- 

the  molecules  of  a  gas  are  not  at  rest,  but     ing 
p 

are  continually  moving  about. 

68.  Molecular  motions  and  the  indefinite  expansibility  of 
a  gas.  Perhaps  the  most  striking  property  which  we  have 
found  gases  to  possess  is  the  property  of  indefinite  or  unlim- 
ited expansibility.  The  existence  of  this  property  was  demon- 
strated by  the  fact  that  we  were  able  to  obtain  a  high  degree 
of  exhaustion  by  means  of  an  air  pump.  No  matter  how  much 
air  was  removed  from  the  bell  jar,  the  remainder  at  once 
expanded  and  filled  the  entire  vessel.  In  fact,  it  was  only 
because  of  this  property  that  the  air  pump  was  able  to  perform 
its  functions  at  all. 

In  order  to  explain  these  facts  it  used  to  be  assumed  that 
the  molecules  of  gases  exert  mutual  repulsion  upon  one 
another.  This  theory  has  now,  however,  been  completely 
abandoned,  for  it  has  been  conclusively  shown  that  no  such 
repulsions  exist.  The  motions  of  the  molecules  alone  furnish 
a  thoroughly  satisfactory  explanation  of  the  phenomenon.  As 
soon  as  the  piston  of  the  air  pump  is  drawn  up,  some  of  the 
molecules  follow  it  because  they  were  already  moving  in  that 

t 


54  MOLECULAK  MOTIONS 

light  body  is  more  easily  moved  than  a  heavy  one,  the  second 
because  a  body  once  set  in  motion  is  more  quickly  stopped  by 
a  dense  gas  than  by  a  very  rare  one.  At  a  pressure  of  y-^ 
atmosphere  the  dancing  motions  of  small  drops  is  exceedingly 
striking.  There  can  be  no  doubt,  then,  that  what  the  oil  drops 
are  here  seen  to  be  doing,  the  molecules  themselves  are  also 
doing,  only  in  a  much  more  lively  way. 

72.  Molecular  velocities.   From  the  known  weight  of  a  cubic 
centimeter  of  air  under  normal  conditions,  and  the  known  force 
which  it  exerts  per  square  centimeter,  — namely,  1033  grams, 

-  it  is  possible  to  calculate  the  velocity  which  its  molecules 
must  possess  in  order  that  they  may  produce  by  their  collisions 
against  the  walls  this  amount  of  force.  Further,  since  a  cubic 
centimeter  of  hydrogen  which  is  in  condition  to  exert  the  same 
pressure  as  a  cubic  centimeter  of  air  weighs  only  one  four- 
teenth as  much  as  the  air,  it  is  evident  that  the  hydrogen 
molecules  must  be  moving  much  more  rapidly  than  the  air 
molecules,  or  else  they  could  not  exert  the  same  pressure. 
The  result  of  the  calculation  gives  to  the  air  molecules  under 
normal  conditions  a  velocity  of  about  445  meters  per  second, 
while  it  assigns  to  the  hydrogen  molecules  the  enormous 
speed  of  1700  meters  (a  mile)  per  second.  The  speed  of  a 
cannon  ball  is  seldom  greater  than  800  meters  (2500  feet) 
per  second.  It  is  easy  to  see  then,  since  the  molecules  of 
gases  are  endowed  with  such  speeds,  why  air,  for  example, 
expands  instantly  into  the  space  left  behind  by  the  rising 
piston  of  the  air  pump,  and  why  any  gas  always  fills  com- 
pletely the  vessel  which  contains  it. 

73.  Diffusion  of  gases  through  porous  walls.    Strong  evi- 
dence for  the  correctness  of  the  above  views  is  furnished  by 
the  following  experiment: 

Let  a  porous  cup  of  unglazed  earthenware  be  closed  with  a  rubber 
stopper  through  which  a  glass  tube  passes,  as  in  Fig.  60.  Let  the  tube 
be  dipped  into'  a  dish  of  colored  water,  and  a  jar  containing  hydrogen 


JAMES  CLERK-MAXWELL 

(1831-1879) 

One  of  the  greatest  of  mathemati- 
cal physicists  ;  born  in  Edinburgh, 
Scotland ;  professor  of  natural 
philosophy  at  Marischal  College, 
Aberdeen,  in  1856,  of  physics  and 
astronomy  in  Kings  College,  Lon- 
don, in  1860,  and  of  experimental 
physics  in  Cambridge  University 
from  1871  to  1879  ;  one  of  the  most 
prominent  figures  in  the  develop- 
ment of  the  kinetic  theory  of 
gases  and  the  mechanical  theory 
of  heat ;  author  of  the  electro- 
magnetic theory  of  light  — a  the- 
ory which  has  become  the  basis  of 
nearly  all  modern  theoretical  work 
in  electricity  and  optics  (see  p.  416) 


HEINRICH  RUDOLPH  HERTZ 

(1857-1894) 

One  of  the  most  brilliant  of  Ger- 
man physicists,  who,  in  spite  of  his 
early  death  a,t  the  age  of  thirty- 
seven,  made  notable  contributions 
to  theoretical  physics,  and  left  be- 
hind the  epoch-making  experimen- 
tal discovery  of  the  electromagnetic 
waves  predicted  by  Maxwell.  Wire- 
less telegraphy  is  but  an  applica- 
tion of  this  discovery  of  so-called 
"  Hertzian  "  waves  (see  p.  413).  The 
capital  discovery  that  ultra-violet 
light  discharges  negatively  electri- 
fied bodies  is  also  due  to  Hertz 


KINETIC  THEORY  OF  GASES 


55 


placed  over  the  porous  cup,  or  let  the  jar  simply  be  held  in  the  position 
shown  in  the  figure,  and  illuminating  gas  passed  into  it  by  means  of  a 
rubber  tube  connected  with  a  gas  jet.  The  rapid  passage  of  bubbles  out 
through  the  water  will  show  that  the  gaseous  pressure  inside  the  cup  is 
rapidly  increasing.  Now  let  the  bell  jar  be  lifted,  so  that  the  hydrogen 
is  removed  from  the  outside.  Water  will  at  once 
begin  to  rise  in  the  tube,  showing  that  the  inside 
pressure  is  now  rapidly  decreasing. 

The  explanation  is  as  follows :  We  have 
learned  that  the  molecules  of  hydrogen  have 
about  four  times  the  velocity  of  the  mole- 
cules of  air.  Hence,  if  there  are  as  many 
hydrogen  molecules  per  cubic  centimeter 
outside  the  cup  as  there  are  air  molecules 
per  cubic  centimeter  inside,  the  hydrogen 
molecules  will  strike  the  outside  of  the  wall 
four  times  as  frequently  as  the  air  molecules 
will  strike  the  inside.  Hence,  in  a  given 
time,  the  number  of  hydrogen  molecules 
which  pass  into  the  interior  of  the  cup 
through  the  little  holes  in  the  porous  material  will  be  four 
times  as  great  as  the  number  of  air  particles  which  pass  out. 
Since  the  inside  is  thus  gaining  molecules  faster  than  it  is  los- 
ing them,  and  since  the  pressure  of  a  gas  at  a  given  tempera- 
ture is  determined  solely  by  the  number  of  molecules  which 
are  bombarding  the  wall,  the  inside  pressure  must  increase 
until  the  number  per  cubic  centimeter  inside  is  so  much  larger 
than  the  number  outside  that  molecules  pass  out  as  fast  as 
they  pass  in.  When  the  bell  jar  is  removed  the  hydrogen 
which  has  passed  inside  now  begins  to  pass  out  faster  than 
the  outside  air  passes  in,  and  hence  the  inside  pressure  is 
diminished. 

74.  Temperature  and  molecular  velocity.  The  effects  which 
are  observed  when  a  gas  is  heated  furnish  further  evidence 
that  its  molecules  are  in  motion. 


FIG.  60.  Diffusion  of 

hydrogen      through 

porous  cup 


56  MOLECULAR  MOTIONS 

Let  a  bulb  of  air  B  be  connected  with  a  water  manometer  m,  as  in 
Fig.  61.    If  the  bulb  is  warmed  by  holding  a  Bunsen  burner  beneath  it, 
or  even  by  placing  the  hand  upon  it,  the  water  at  in  will,  at  once  begin 
to  descend,  showing  that  the  pressure  exerted  by 
the  air  contained  in  the  bulb  has  been  increased 
by  the  increase  in  its  temperature.    If  B  is  cooled 
with  ice  or  ether,  the  water  will  rise  at  m. 


Now  if  gas  pressure  is  due  to  the  bom- 
bardment of  the  walls  by  the  molecules  of 
the  gas,  since  the  number  of  molecules  in 
the  bulb  can  scarcely  have  been  changed 
by  slightly  heating  it,  we  are  forced  to 
conclude  that  the  increase  in  pressure  is 
due  to  an  increase  in  the  velocity  of  the 

molecules   which   are    already   there.     The 

.  PI       FIG.  61.    Expansion 

temperature  of  a  given  gas,  then,  from  the        of  air  by  heat 

standpoint  of  the  kinetic  theory,  is  deter- 
mined simply  by  the  mean  velocity  of  the  gas  molecules.  To 
increase  the  temperature  is  to  increase  the  average  velocity  of 
the  molecules,  and  to  diminish  the  temperature  is  to  diminish 
this  average  molecular  velocity.  The  theory  thus  furnishes  a 
very  simple  and  natural  explanation  of  the  fact  of  the  expan- 
sion of  gases  with  a  rise  in  temperature. 


QUESTIONS  AND  PROBLEMS 

1.  Automobile  tires  are  pumped  up  to  a  pressure  of  80  Ib.  per  sq.  in. 
What  is  the  density  of  the  contained  air  ?    (1  atmosphere  =  14.7  lb:) 

2.  If  a  vessel  containing  a  small  leak  is  filled  with  hydrogen  at  a 
pressure  of  2  atmospheres,  the  pressure  falls  to  1  atmosphere  about 
4  times  as  fast  as  when  the  same  experiment  is  tried  with  air.    Can 
you  see  a  reason  for  this  ? 

3.  A  liter  of  air  at  a  pressure  of  76  cm.  is  compressed  so  as  to  occupy 
400  cc.    What  is  the  pressure  against  the  walls  of  the  containing  vessel ? 

4.  If  an  open  vessel  contains  250  g.  of  air  when  the  barometric  height 
is  750  mm.,  what  weight  will  the  same  vessel  contain  at  the  same  tem- 
perature when  the  barometric  height  is  740  mm.  ? 


MOLECULAR  MOTIONS  IN  LIQUIDS  57 

5.  The  density  of  air  is  .001293  when  the  temperature  is  0°  C.  and 
the  pressure  76  cm.    How  large  must  a  vessel  be  to  contain  a  kilogram 
of  air  when  the  temperature  is  0°  C.  and  the  pressure  75  cm.  ? 

6.  On  a  day  on  which  the  barometric  height  is  76  cm.  the  volume  of 
the  space  above  the  mercury  in  a  Torricelli  tube  is  10  cc.,  and  on  account 
of  air  in  this  space  the  mercury  in  the  tube  stands  only  74  cm.  high. 
How  high  will  the  mercury  stand  above  the  cistern  if  the  tube  is  pulled 
up  out  of  the  dish  so  that  the*  space  above  is  20  cc.  ? 

7.  Find  the  pressure  to  which  the  diver  was  subjected  who  descended 
to  a  depth  of -201  ft.   Find  the  density  of  the  air  in  his  suit,  the  density 
at  the  surface  being  .00118  and  the  temperature  being  assumed  to  remain 
constant.'  Take  the  pressure  at  the  surface  as  75  cm. 

8.  A  bubble  of  air  which  escaped  from  this  diver's  suit  would  increase 
to  how  many  times  its  volume  on-reaching  the  surface? 


MOLECULAR  MOTIONS  IN  LIQUIDS 

75.  Molecular  motions  in  liquids  and  evaporation.  Evidence 
that  the  molecules  of  liquids  as  well  as  those  of  gases  are  in  a 
state  of  perpetual  motion  is  found,  first,  in  the  familiar  facts 
of  evaporation. 

We  know  that  the  molecules  of  a  liquid  in  an  open  vessel 
are  continually  passing  off  into  the  space  above ;  for  it  is  only 
a  matter  of  time  when  the  liquid  completely  disappears  and  the 
vessel  becomes  dry.  Now  it  is  hard  to  imagine  a  way  in  which 
the  molecules  of  a  liquid  thus  pass  out  of  the  liquid  into  the 
space  above,  unless  these  molecules,  while  in  the  liquid  condition, 
are  in  motion.  As  soon,  however,  as  such  a  motion  is  assumed, 
the  facts  of  evaporation  become  perfectly  intelligible.  For  it  is 
to  be  expected  that  in  the  jostlings  and  collisions  of  rapidly 
moving  liquid  molecules  an  occasional  molecule  will  acquire  a 
velocity  much  greater  than  the  average.  This  molecule  may 
then,  because  of  the  unusual  speed  of  its  motion,  break  away 
from  the  attraction  of  its  neighbors  and  fly  off  into  the  space 
above.  This  is  indeed  the  mechanism  by  which  we  now  believe 
that  the  process  of  evaporation  goes  on  from  the  surface  of 
any  liquid. 


58  MOLECULAR  MOTIONS 

76.  Molecular  motions  and  the  diffusion  of  liquids.    One  of 

the  most  convincing  arguments  for  the  motions  of  molecules 
in  gases  was  found  in  the  fact  of  diffusion.  But  precisely  the 
same  sort  of  phenomena  are  observable  in  liquids. 

Let  a  few  lumps  of  blue  litmus  be  pulverized  and  dissolved  in  water. 
Let  a  tall  glass  cylinder  be  half  filled  with  this  water  and  a  few  drops 
of  ammonia  added.    Let  the  remainder  of  the  litmus  solution  be  turned 
red  by  the  addition  of  one  or  two  cubic  centimeters 
of  nitric  acid.     Then  let  this  acidulated  water  be  f^ 

introduced  into  the  bottom  of  the  jar  through  a 
thistle  tube  (Fig.  62).  In  a  few  minutes  the  line  of 
separation  between  the  acidulated  water  and  the 
blue  solution  will  be  fairly  sharp ;  but  in  the  course 
of  a  few  hours,  even  though  the  jar  is  kept  perfectly 
quiet,  the  red  color  will  be  found  to  have  spread 
considerably  toward  the  top  of  the  jar,  showing  that 
the  acid  molecules  have  gradually  found  their  way 
toward  the  top. 

FIG.  62.   Diffusion 

Certainly,  then,  the  molecules  of  a  liquid         Of  liquids 
must  be  endowed  with  the  power  of  independ- 
ent motion.    Indeed,  every  one  of  the  arguments  for  molec- 
ular motions   in  gases   applies  with  equal  force  to  liquids. 
Even  the  Brownian  movements  can  be  seen  in  liquids,  though 
they  are  here  so  small  that  high  power  microscopes  must  be 
used  to  make  them  apparent. 

77.  Molecular  motions  and  the  expansion  of  liquids.  The  fact 
of  the  expansion  of  gases  with  a  rise  of  temperature  was  looked 
upon  as  evidence  that  the  molecules  of  gases  are  in  motion,  the 
velocity  of  this  motion  increasing  with  an  increase  in  tempera- 
ture.   But  precisely  the  same  property  belongs  to  liquids  also. 

Thus,  let  the  bulb  (Fig.  63)  be  heated  with  a  Bunsen  burner.  The 
contained  liquid  will  be  found  to  expand  and  rise  in  the  tube. 

It  is  natural  to  infer  that  the  cause  of  this  increase  in  volume 
is  the  same  as  before ;  that  is,  the  velocity  of  the  molecules 
of  the  liquid  has  been  increased  by  the  rise  in  temperature, 


PROPERTIES  OF  VAPORS 


59 


and  they  have  therefore  jostled  one  another  farther  apart,  and 
thus  caused  the  whole  volume  to  be  enlarged.    According  to 
this  view,  then,  an  increase  in  temperature  in  a  liquid,  as  in  a 
gas,  means  an  increase  in  the  mean  velocity  of  the 
molecules,  and  conversely  a  decrease  in  temper- 
ature means  a  decrease  in  this  average  velocity. 
78.  Evaporation  and  temperature.    If  it  is  true 
that   increase  in  temperature  means  increase  in 
the  mean  velocity  of  molecular  motion,  then  the 
number  of  molecules  which  chance   in   a  given 
time  to  acquire  the  velocity  necessary  to  carry 
them  into  the   space   above   the  liquid  ought  to 
increase   as   the  temperature   increases ;    that   is, 
evaporation  ought  to  take  place  more  rapidly  at 
high  temperatures  than  at  low.    Common  obser- 
vation  teaches  that  this  is  true.     Damp  clothes    pansion  of  a 
become  dry  under  a  hot  flatiron  but  not  under         liquid 
a    cold    one;    the   sidewalk    dries    more    readily 
in   the   sun    than    in    the    shade ;   we   put  wet   objects  near 
a  hot  stove  or  radiator  when  we  wish  them  to  dry  quickly. 


PROPERTIES  OF  VAPOR'S 

79.  Saturated  vapor.  If  a  liquid  is  placed  in  an  open  vessel, 
there  ought  to  be  no  limit  to  the  number  of  molecules  which 
can  be  lost  by  evaporation,  for  as  fast  as  the  molecules  emerge 
from  the  liquid  they  are  carried  away  by  air  currents.  As  a 
matter  of  fact,  experience  teaches  that  water  left  in  an  open 
dish  does  waste  away  until  the  dish  is  completely  dry. 

But  suppose  that  the  liquid  is  evaporating  into  a  closed 
space,  such  as  that  shown  in  Fig.  64.  Since  the  molecules 
which  leave  the  liquid  cannot  escape  from  the  space  S,  it  is 
clear  that  as  time  goes  on  the  number  of  molecules  which  have 
passed  off  from  the  liquid  into  this  space  must  continually 


60  MOLECULAE  MOTIONS 

increase ;  in  other  words,  the  density  of  the  vapor  in  S  must 
grow  greater  arid  greater.  But  there  is  an  absolutely  definite 
limit  to  the  density  which  the  vapor  can  attain.  For,  as  soon 
as  it  reaches  a  certain  value,  depending  on  the  temperature 
and  on  the  nature  of  the  liquid,  the  number  of  molecules 
returning  per  second  to  the  liquid  surface  will  be  exactly 
equal  to  the  number  escaping.  The  vapor  is 
then  said  to  be  saturated.  ff  ^^ 

If  the  density  of  the  vapor  is  lessened  tem- 
porarily by  increasing  the  size  of  the  vessel  S, 
more  molecules  will  escape  from  the  liquid  per 

second  than  return  to  it  until  the  density  of 

,  .       ,    .,          •    •      -T       i  FIG.  64.  Asatu- 

the  vapor  has  regained  its  original  value.  rated  vapor 

If,  on  the  other  hand,  the  density  of  the  vapor 
has  been  increased  by  compressing  it,  more  molecules  return  to 
the  liquid  per  second  than  escape,  and  the  density  of  the  vapor 
falls  quickly  to  its  "  saturated  "  value.  We  learn,  then,  that  the 
density  of  the  saturated  vapor  of  a  liquid  depends  on  the  tempera- 
ture alone,  and  cannot  be  affected  by  changes  in  volume. 

80.  Pressure  of  a  saturated  vapor.  Just  as  a  gas  exerts  a 
pressure  against  the  walls  of  the  containing  vessel  by  the 
blows  of  its  moving  molecules,  so  also  does  a  confined  vapor. 
But  at  any  given  temperature  the  density  of  a  saturated  vapor 
can  have  only  a  definite  value,  that  is,  there  can  be  only  a 
definite  number  of  molecules  per  cubic  centimeter.  It  fol- 
lows, therefore,  that  just  as  at  any  temperature  the  saturated 
vapor  can  have  only  one  density,  so  also  it  can  have  only  one 
pressure.  This  pressure  is  called  the  pressure  of  the  saturated 
vapor  corresponding  to  the  given  temperature. 

Let  four  Torricelli  tubes  be  set  up  as  in  Fig.  65,  and  with  the  aid  of 
a  curved  pipette  (Fig.  65)  let  a  drop  of  ether  be  introduced  into  the 
bottom  of  tube  1.  This  drop  will  at  once  rise  to  the  top  and  a  portion 
of  it  will  evaporate  into  the  vacuum  which  exists  above  the  mercury. 
The  pressure  of  this  vapor  will  push  down  the  mercury  column,  and  the 


PROPERTIES  OF  VAPORS 


61 


number  of  centimeters  of  this  depression  will  of  course  be  a  measure  of 
the  pressure  of  the  vapor.  It  will  be  observed  that  the  mercury  will  fall 
almost  instantly  to  the  lowest  level  which  it  will  ever  reach  —  a  fact 
which  indicates  tl^t  it  takes  but  a  vefy  short  time  for  the  condition  of 
saturation  to  be  attained.  In  the  same  way  let  alcohol  and  water  be 
introduced  into  tubes  2  and  3  respectively. 

While  the  pressure  of  the  saturated  ether  vapor  at  the 
temperature  of  the  room  will  be  found  to  be  as  much  as  40 
centimeters,  that  of  alcohol  will  be  found  to  be  but  4  or 
5  centimeters,  and  that  of  water 
only  1  or  2  centimeters. 


4-  3   2  1 


Let  a  Bunsen  flame  be  passed  quickly 
across  the  tubes  of  Fig.  65  near  the  upper 
level  of  the  mercury.  The  vapor  pressure 
will  increase  rapidly  in  all  of  the  tubes,  as 
shown  by  the  fall  of  the  mercury  columns. 
This  will  be  especially  noticeable  in  the 
case  of  the  ether. 

The  experiment  proves  that  both 
the  pressure  and  the  density  of  a 
saturated  vapor  increase  rapidly 
with  the  temperature.  This  was 
to  have  been  expected  from  our 

theory;  for  increasing  the  temper-    FlG'  65'    VaP°/  Pressures  of 

0  saturated   vapors 

ature   of   the   liquid   increases  the 

mean  velocity  of  its  molecules  and  hence  increases  the  num- 
ber which  attain  each  second  the  velocity  necessary  for  escape. 

Let  air  be  introduced  into  tube  4  until  the  mercury  stands  at  about 
the  same  height  as  in  tube  1.  Let  pieces  of  ice  be  held  against  tubes  1 
and  4  near  the  top  of  the  mercury.  The  mercury  will  rise  in  both,  but 
much  more  rapidly  in  the  ether  tube  than  in  the  air  tube,  thus  showing 
that  the  ether  vapor  is  condensing. 

The  experiment  shows  that  if  the  temperature  of  a  saturated 
vapor  is  diminished,  it  condenses  until  its  density  is  reduced 


MOLECULAR  MOTIONS 


to  that  corresponding  to  saturation  at  the  lower  temperature. 
How  rapidly  the  density  and  pressure  of  saturation  increase 
with  temperature  may  be  seen  from  the  following  table : 

TABLE  OF  CONSTANTS  OF  SATURATED  WATER  VAPOR 

The  table  shows  the  pressure  P,  in  millimeters  of  mercury,  and  the  den- 
sity D  of  aqueous  vapor  saturated  at  temperatures  t°  C. 


t. 

P. 

D. 

t. 

P. 

I). 

t. 

P. 

D. 

-  10° 

2.2 

.0000023 

4° 

6.1 

.0000064 

18° 

15.3 

.0000152 

-  9° 

2.3 

.0000025 

5° 

6.5 

.0000068 

19° 

16.3 

.0000162 

-  8° 

2.5 

.0000027 

6° 

7.0 

.0000073 

20° 

17.4 

.0000172 

-  7° 

2.7 

.0000029 

7° 

7.5 

.0000077 

21° 

18.5 

.0000182 

-  6° 

2.9 

.0000032 

8° 

8.0 

.0000082 

22° 

19.6 

.0000193 

-  5° 

3.2 

.0000034 

9° 

8.5 

.0000087 

23° 

20.9 

.0000204 

-  4° 

3.4 

.0000037 

10° 

9.1 

.0000093 

24° 

22.2 

.0000216 

-  3° 

3.7 

.0000040 

11° 

9.8 

.0000100 

25° 

23.5 

.0000229 

_  2° 

3.9 

.0000042 

12° 

10.4 

.0000106 

26° 

25.0 

.0000242 

-  1° 

4.2 

.0000045 

13° 

11.1  . 

.0000112 

27° 

26.5 

.0000256 

0° 

4.6 

.0000049 

14° 

11.9 

.0000120 

28° 

28.1 

.0000270 

1° 

4.9 

.0000052 

15° 

12.7 

.0000128 

30° 

31.5 

.0000301 

2° 

5.3 

.0000056 

16° 

13.5 

.0000135 

35° 

41.8 

.0000393 

3° 

5.7 

.0000060 

17° 

14.4 

.0000144 

40° 

54.9 

.0000509 

81.  The  influence  of  air  on  evaporation.  We  observed  that 
when  a  drop  of  ether  was  inserted  into  a  Torricelli  tube  the  mer- 
cury fell  very  suddenly  to  its  final  position,  showing  that  in  a 
vacuum  the  condition  of  saturation  is  reached  almost  instantly. 
This  was  to  have  been  expected  from  the  great  velocities  which 
we  found  the  molecules  of  gases  and  vapors  to  possess. 

In  order  to  see  what  effect  the  presence  of  air  has  upon  evaporation, 
let  a  drop  of  ether  be  introduced  into  a  Torricelli  tube  which  is  partly 
filled  with  air.  The  mercury  will  not  now  be  found  to  sink  instantly  to 
its  final  level  as  it  did  before,  but  although  it  will  fall  rapidly  at  first, 
it  will  continue  to  fall  slowly  for  several  hours.  At  the  end  of  a  day,  if 
the  temperature  has  remained  constant,  it  will  show  a  depression  which 
indicates  a  vapor  pressure  of  the  ether  just  as  great  as  that  existing  in 
a  tube  which  contains  no  air. 


PROPERTIES  OF  VAPORS  63 

The  experiment  leads,  then,  to  the  rather  remarkable  con- 
clusion tli&tjust  as  much  liquid  will  evaporate  into  a  space  which 
is  already  full  of  air  as  into  a  vacuum.  The  air  has  no  effect 
except  to  retard  greatly  the  rate  of  evaporation. 

82.  Explanation  of  the  retarding  influence  of  air  on  evapo- 
ration. This  retarding  influence  of  air  on  evaporation  is  easily 
explained  by  the  kinetic  theory;  for  while  in  a  vacuum  the 
molecules  which  emerge  from  the  surface  fly  at  once  to  the 
top  of  the  vessel,  when  air  is  present  the  escaping  molecules 
collide  with  the  air  molecules  before  they  have  gone  any  appre- 
ciable distance  away  from  the  surface  (probably  less  than 
.00001  centimeter),  and  only  work  their  way  up  to  the  top 
after  an  almost  infinite  number  of  collisions.  Thus,  while  the 
space  immediately  above  the  liquid  may  become  saturated  very 
quickly,  it  requires  a  long  time  for  this  condition  of  saturation 
to  reach  the  top  of  the  vessel. 

It  must  not  be  forgotten,  however,  that  at  a  given  tempera- 
ture the  pressure  existing  within  a  vessel  containing  gases  is 
simply  due  to  the  total  number  of  molecules  per  cubic  centi- 
meter which  are  striking  blows  against  each  square  centimeter 
of  the  wall.  Therefore,  when  a  liquid  evaporates  into  a  closed 
vessel  already  containing  air,  the  pressure  gradually  increases, 
and  is  ultimately  equal  to  the  air  pressure  plus  the  pressure  of 
the  saturated  vapor.  When  a  liquid  evaporates  in  an  open  ves- 
sel, —  that  is,  under  constant  pressure,  —  its  molecules  crowd 
out  an  equal  number  of  molecules  of  air. 


QUESTIONS  AND  PROBLEMS 

1.  Salt  is  heavier  than  water.    Why  does  not  all  the  salt  in  a  mix- 
ture of  salt  and  water  settle  to  the  bottom  ? 

2.  The  space  above  the  mercury  in  a  Torricelli  is  filled  with  satu- 
rated ether  vapor.    Its  volume  is  20  cc.  and  the  height  of  the  mercury 
is  36  cm.    The  tube  is  pushed  down  into  the  mercury  cistern  until  the 
volume  occupied  by  the  vapor  is  10  cc.    What  is  now  the  height  of  the 
mercury  ? 


64  MOLECULAR  MOTIONS 

3.  If  the  inside  of  a  barometer  tube  is  wet  when  it  is  filled  with  mer- 
cury, will  the  height  of  the  mercury  be  the  same  as  in  a  dry  tube  ? 

4.  At  a  temperature  of  15°  C.  what  will  be  the  error  in  the  baro- 
metric height  indicated  by  a  barometer  which  contains  moisture  ?   (See 
Table  of  Constants  of  Saturated  Water  Vapor,  p.  62.) 

5.  Why  do  clothes  dry  more  quickly  on  a  windy  day  than  on  a  quiet 
day? 

6.  If  dry  air  were  placed  in  a  closed  vessel  when  the  barometer  was 
76  cm.,  and  a  dish  of  water  then  introduced  within  the  closed  space, 
what  pressure  would  finally  be  attained  within  the  vessel  if  the  temper- 
ature were  kept  at  18°  C.  ? 

7.  How  many  grams  of  water  will  evaporate  at  20°  C.  into  a  closed 
room  18  x  20  x  4  in.  ?    (See  table,  p.  62,  for  density  of  saturated  water 
vapor  at  20°  C.) 

HYGROMETRY,  OR  THE  STUDY  OF  MOISTURE  CONDITIONS 
IN  THE  ATMOSPHERE  * 

83.  Condensation  of  water  vapor  from  the  air.  Were  it  not 
for  the  retarding  influence  of  air  upon  evaporation  we  should 
be  obliged  to  live  in  an  atmosphere  which  would  be  always 
completely  saturated  with  water  vapor;  for  the  evaporation 
from  oceans,  lakes,  and  rivers  would  almost  instantly  saturate 
all  the  regions  of  the  earth.  This  condition  —  one  in  which 
moist  clothes  would  never  dry,  and  in  which  all  objects  would 
be  perpetually  soaked  in  moisture  —  would  be  exceedingly 
uncomfortable,  if  not  altogether  unendurable. 

But  on  account  of  the  slowness  with  which,  as  the  last  ex- 
periment showed,  evaporation  takes,  place  into  air,  the  water 
vapor  which  always  exists  in  the  atmosphere  is  usually  far 
from  saturated,  even  in  the  immediate  neighborhood  of  lakes 
and  rivers.  Since,  however,  the  amount  of  vapor  which  is 
necessary  to  produce  saturation  rapidly  decreases  with  a  fall 
in  temperature,  if  the  temperature  decreases  continually  in 
some  unsaturated  locality,  it  is  clear  that  a  point  must  soon 

*  It  is  recommended  that  this  subject  be  preceded  by  a  laboratory  determina- 
tion of  dew  point,  humidity,  etc.  See,  for  example,  Experiment  10  of  the  authors' 
manual. 


HYGROMETRY  65 

be  reached  at  which  the  amount  of  vapor  already  existing  in  a 
cubic  centimeter  of  the  atmosphere  is  the  amount  correspond- 
ing to  saturation.  Then,  in  accordance  with  the  facts  discov- 
ered in  §  80,  if  the  temperature  still  continues  to  fall,  the 
vapor  must  begin  to  condense.  Whether  it  condenses  as  dew, 
or  cloud,  pr  fog,  or  rain  will  depend  upon  how  and  where  the 
cooling  takes  place. 

84.  The  formation  of  dew.   If  the  cooling  is  due  to  the  natu- 
ral radiation  of  heat  from  the  earth  at  night  after  the  sun's 
warmth  is  withdrawn,  the  atmosphere  itself  does  not  fall  in 
temperature  nearly  as  rapidly  as  do  solid  objects,  on  the  earth, 
such  as  blades  of  grass,  trees,  stones,  etc.    The  layers  of  air 
which  come  into  immediate  contact  with  these  cooled  bodies 
are  themselves  cooled,  and  as  they  thus  reach  a  temperature 
at  which  the  amount  of  moisture  which  they  already  contain 
is  in  a  saturated  condition,  they  begin  to  deposit  this  mois- 
ture, in  the  form  of  dew,  upon  the  cold  objects.   The  drops  of 
moisture  which  collect  on  an  ice  pitcher  in  summer  illustrate 
perfectly  the  whole  process. 

85.  The  formation  of  fog.    If  the  cooling  at  night  is  so 
great  as  not  only  to  bring  the  grass  and  trees  below  the  tem- 
perature at  which  the  vapor  in  the  air  in  contact  with  them  is 
in  a  state  of  saturation,  but  also  to  lower  the  whole  body  of 
air  near  the  earth  below  this  temperature,  then  the  condensa- 
tion takes  place  not ''only  on  the  solid  objects  but  also  on  dust 
particles  suspended  in  the  atmosphere.    This  constitutes  a  fog. 

86.  The  formation  of  clouds,  rain,  hail,  and  snow.    When 
the  cooling  of  the  atmosphere  takes  place  at  some  distance 
above  the  earth's  surface,  as  when  a  warm  current  of  ah-  enters 
a  cold  region,  if  the  resultant  temperature  is  below  that  at 
which  the  amount  of  moisture  already  in  the  air  is  sufficient 
to  produce  saturation,  this   excessive  moisture  immediately 
condenses  .about  floating  dust  particles  and  forms  a  cloud. 
If  the  cooling  is  sufficient  to  free  a  considerable  amount  of 


66  MOLECULAR  MOTIONS 

moisture,  the  drops  become  large  and  fall  as  rain.  If  this 
falling  rain  passes  through  cold  regions,  it  freezes  into  hail. 
If  the  temperature  at  which  condensation  begins  is  below 
freezing,  the  condensing  moisture  forms  into  snow-flakes. 

87.  The  dew  point.    The  temperature  to  which  the  atmos- 
phere must  be  cooled  in  order  that  condensation  may  begin 
is  called  the  dew  point.     This  temperature  may 

be  found  by  partly  filling  with  water  a  brightly 
polished  vessel  of  200  or  300  cubic  centimeters 
capacity  and  dropping  into  it  little  pieces  of  ice, 
stirring  thoroughly  at  the  same  time  with  a  ther- 
mometer.    The   dew   point   is   the 
temperature  indicated  by  the  ther- 
mometer at  the  instant  a  film  of 
moisture    appears    upon    the    pol- 
ished surface.     In  winter  the  dew 
point    is    usually    below    freezing,    FIG.  66.    Apparatus  for  deter- 
and  it  will  therefore  be  necessary 

to  add  salt  to  the  ice  and  water  in  order  to  make  the  film 
appear.  The  experiment  may  be  performed  equally  well 
by  bubbling  a  current  of  air  through  ether  contained  in  a 
polished  tube  (Fig.  66). 

88.  Humidity  of  the  atmosphere.    From  the  clew  point  and 
table  given  in  §  80,  p.  62,  we  can  easily  find  what  is  com- 
monly known  as  the  relative  humidity,  or  the  degree  of  satura- 
tion of  the  atmosphere.    This  quantity  is  defined  as  the  ratio 
between  the  amount  of  moisture  actually  present  in  the  air  per 
cubic  centimeter  and  the  amount  which  would  be  present  if  the 
air  were  completely  saturated.    This  is  precisely  the  same  as  the 
ratio  between  the  pressure  which  the  water  vapor  present  in 
the  air  exerts  and  the  pressure  which  it  would  exert  if  it 
were  present  in  sufficient  quantity  to  be  in  the  saturated  con- 
dition.   An  example  will  make  clear  the  method  of  finding 
the  relative  humidity. 


HYGROMETRY  67 

Suppose  that  the  dew  point  were  found  to  be  15°  C.  on  a  day  on  which 
the  temperature  of  the  room  was  25°  C.  The  amount  of  moisture  actu- 
ally present  in  the  air  then  saturates  it  at  15°  C.  We  see  from  the  P 
column  in  the  table  that  the  pressure  of  saturated  vapor  at  15°  C.  is 
12.7  millimeters.  This  is,  then,  the  pressure  exerted  by  the  vapor  in 
the  air  at  the  time  of  our  experiment.  Running  down  the  table,  we 
see  that  the  amount  of  moisture  required  to  produce  saturation  at  the 
temperature  of  the  room,  that  is,  at  25°,  would  exert  a  pressure  of 
23.5  millimeters.  Hence  at  the  time  of  the  experiment  the  air  con- 
tains 12.7/23.5,  or  .54,  as  much  water  vapor  as  it  might  hold. 
We  say,  therefore,  that  the  air  is  54%  saturated,  or  that  the  relative 
humidity  is  54%. 

89.  Practical  value  of  humidity  determinations.    From  hu- 
midity determinations  it  is  possible  to  obtain  much  information 
regarding  the  likelihood  of  rain  or  frost.    Such  observations 
are  continually  made  for  this  purpose  at  all  meteorological 
stations.    Further,  they  are  made  in  greenhouses  to  see  that 
the  air  does  not  become  too  dry  for  the  welfare  of  the  plants, 
and  also  in  hospitals  and  public  buildings,  and  even  in  private 
dwellings,  in  order  to  Insure  the  maintenance  of  hygienic  liv- 
ing conditions.    For  the  most  healthful  conditions  the  relative 
humidity  should  be  kept  at.  from  50%  to  60%. 

90.  Cooling  effect  of  evaporation.    Let  three  shallow  dishes  be 
partly  filled,  the  first  with  water,  the  second  with  alcohol,  and  the  third 
with  ether,  the  bottles  from  which  these  liquids  are  obtained  having  stood 
in  the  room  long  enough  to  acquire  its  temperature.    Let  three  students 
carefully  read  as  many  thermometers,  first  before  their  bulbs  have  been 
immersed  in  the  respective  liquids  and  then  after.    In  every  case  the 
temperature  of  the  liquid  in  the  shallow  vessel  will  be  found  to  be 
somewhat  lower  than  the  temperature  of  the  air,  the  difference  being 
greatest  in  the  case  of  ether  and  least  in  the  case  of  water. 

It  appears  from  this  experiment  that  an  evaporating  liquid 
assumes  a  temperature  somewhat  lower  than  its  surroundings, 
and  that  the  substances  which  evaporate  the  most  readily,  that 
is,  those  which  have  the  greatest  vapor  pressures  at  a  given 
temperature  (see  §  80),  assume  the  lowest  temperatures. 

t 


68  MOLECULAR  MOTIONS 

• 

Another  way  of  establishing  the  same  truth  is  to  place  a  few  drops 
of  each  of  the  above  liquids  in  succession  on  the  bulb  of  the  arrange- 
ment shown  in  Fig.  61,  and  observe  the  rise  of  water  in  the  stem ;  or, 
more  simply  still,  to  place  a  few  drops  of  each  liquid  on  the  back  of  the 
hand,  and  notice  that  the  order  in  which  they  evaporate  —  namely, 
ether,  alcohol,  water — is  the  order  of  greatest  cooling. 

91.  Explanation  of  the  cooling  effect  of  evaporation.    The 
kinetic  theory  furnishes  a  simple  explanation  of  the  cooling 
effects  of  evaporation.    We  saw  that  in  accordance  with  this 
theory  evaporation  means  an  escape  from  the  surface  of  those 
molecules  which  have  acquired  velocities  considerably  above 
the  average.    But  such  a  continual  loss  from  a  liquid  of  its 
most  rapidly  moving  molecules  involves,  of  course,  a  continual 
diminution    of    the   average    velocity   of    the  molecules  left 
behind  in  the  liquid  state,  and  this  means  a  decrease  in  the 
temperature  of  the  liquid  (see  §§74  and  77). 

Again,  we  should  expect  the  amount  of  cooling  to  be  pro- 
portional to  the  rate  at  which  the  liquid  is  losing  molecules. 
Hence,  of  the  three  liquids  studied,  ether  should  cool  most 
rapidly,  since  it  shows  the  highest  vapor  pressure  at  a  given 
temperature  and  therefore  the  highest  rate  of  emission  of 
molecules.  The  alcohol  should  come  next  in  order,  and  the 
water  last,  as  was  in  fact  observed. 

92.  Freezing  by  evaporation.    In  §  81  it  was  shown  that  a 
liquid  will  evaporate  much  more  quickly  into  a  vacuum  than 
into  a  space  containing  air.    Hence  if  we  place  a  liquid  under 
the  receiver  of  an  air  pump  and  exhaust  rapidly,  we  ought  to 
expect  a  much  greater  fall  in   temperature  than  when  the 
liquid  evaporates  into  air.    This  conclusion  may  be  strikingly 
verified  as  follows : 

Let  a  thin  watch  glass  be  filled  with  ether  and  placed  upon  a  drop  of 
cold  water,  preferably  ice  water,  which  rests  upon  a  thin  glass  plate. 
Let  the  whole  arrangement  be  placed  underneath  the  receiver  of  an  air 
pump  and  the  air  rapidly  exhausted.  After  a  few  minutes  of  pumping 
the  watch  glass  will  be  found  frozen  to  the  plate. 


HYGEOMETKY 


69 


By  evaporating  liquid  helium  in  this  way  Professor  Kam- 
erlingh  Onnes  of  Leyderi,  in  1911,  attained  the  lowest  tem- 
perature which  has  ever  been  reached,  namely,  —  271.3°  C. 
or  -  456.3°  F. 

93.  Effect  of  air  currents  upon  evaporation.    Let  four  ther- 
mometer bulbs,  the  first  of  which  is  dry,  the  second  wetted  with  water, 
the  third  with  alcohol,  and  the  fourth  with  ether,  be  rapidly  fanned  and 
their  respective  temperatures  observed.    The  results  will  show  that  in 
all  of  the  wetted  thermometers  the  fanning  will  considerably  augment 
the  cooling,  but  the  dry  thermometer  will  be  wholly  unaffected. 

The  reason  that  fanning  thus  facilitates  evaporation,  and 
therefore  cooling,  is  that  it  removes  the  saturated  layers  of 
vapor  which  are  in  immediate  contact  with  the  liquid  and 
replaces  them  by  unsaturated  layers  into 
which  new  evaporation  may  at  once  take 
place.  From  the  behavior  of  the  dry- 
bulb  thermometer,  however,  it  will  be 
seen  that  fanning  produces  cooling  only 
when  it  can  thus  hasten  evaporation. 
A  dry  body  at  the  temperature  of  the 
room  is  not  cooled  in  the  slightest 
degree  by  blowing  a  current  of  air 
across  it. 

94.  The  wet-  and  dry-bulb  hygrometer. 
In    the    wet-    and   dry-bulb    hygrometer 
(Fig.  67)    the    principle    of    cooling   by 
evaporation    finds    a   very   useful    appli- 
cation.    This  instrument  consists  of  two 
thermometers,  the  bulb  of  one  of  which 

is  dry,  while  that  of  the  other  is  kept  continually  moist  by 
a  wick  dipping  into  a  vessel  of  water.  Unless  the  air  is  satu- 
rated the  wet  bulb  indicates  a  lower  temperature  than  the 
dry  one,  for  the  reason  that  evaporation  is  continually  taking 
place  from  its  surface.  How  much  lower  will  depend  on  how 


FIG.  67.  Wet-  and  dry- 
bulb  hygrometer 


70  MOLECULAR  MOTIONS 

rapidly  the  evaporation  proceeds ;  and  this  in  turn  will  depend 
upon  the  relative  humidity  of  the  atmosphere.  Thus  in  a 
completely  saturated  atmosphere  no  evaporation  whatever 
takes  place  at  the  wet  bulb,  and  it  consequently  indicates 
the  same  temperature  as  the  dry.  one.  By  comparing  the 
indications  of  this  instrument  with  those  of  the  dew-poiAt 
hygrometer  (Fig.  66),  tables  have  been  constructed  which 
enable  one  to  determine  at  once  from  the  readings  of  the 
two  thermometers  both  the  relative  humidity  and  the  dew 
point.  On  account  of  their  convenience  instruments  of  this 
sort  are  used  almost  exclusively  in  practical  work.  They 
are  not  very  reliable  unless  the  air  is  made  to  circulate 
about  the  wet  bulb  before  the  reading  is  taken.  In  scien- 
tific work  this  is  always  done. 

95.  Effect  of  increased  surface  upon  evaporation.  Let  a  small 

test  tube  containing  a  few  drops  of  water  be  dipped  into  a  larger  tube,  or 
a  small  glass,  containing  ether,  as  in  Fig.  68,  and  let 
a  current  of  air  be  forced  rapidly  through  the  ether 
with  an  aspirator,  in  the  manner  shown.  The  water 
within  the  tube  will  be  frozen  in  a  few  minutes. 

The  effect   of   passing  bubbles  through 
the  ether  is  simply  to  increase  enormously     FlG  68    Freezing 
the  evaporating  surface,  for  the  ether  mole-    water  by  the  evap- 
cules  which  could  before  escape  only  at  the       oration  of  ether 
upper  surface  can  now  escape  into  the  air  bubbles  as  well. 

96.  Factors  affecting  evaporation.    The  above  results  may 
be  summarized  as  follows:   The  rate  of  evaporation  depends 
(1)   on  the  nature  of  the  evaporating  liquid;    (2)  on  the 
temperature  of  the  evaporating  liquid;  (3)  on  the  degree  of 
saturation  of  the  space  into  which  the  evaporation  takes  place ; 
(4)  on  the  density  of  the  air  or  other  gas  above  the  evaporating 
surface  ;  (5)  on  the  rapidity  of  the  circulation  of  the  air  above 
the  evaporating  surface ;    (6)  on  the  extent  of  the  exposed 
surface  of  the  liquid. 


MOLECULAR  MOTIONS  IK  SOLIDS  71 

MOLECULAR  MOTIONS  IN  SOLIDS 

97.  Evidence  for  molecular  motions  in  solids.  We  have  in- 
ferred that  the  molecules  of  liquids  are  in  motion,  in  part  at 
least,  from  the  fact  that  liquids  increase  in  volume  when  the 
temperature  is  raised,  and  from  the  fact  that  molecules  of  the 
liquid  can  usually  be  detected  in  a  gaseous  condition  above 
the  surface.  Both  of  these  reasons  apply  just  as  well  in  the 
case  of  solids. 

Thus  the  facts  of  expansion  of  solids  with  an  increase  in 
temperature  may  be  seen  on  every  side.  Railroad  rails  are 
laid  with  spaces  between  their  ends  so  that 
they  may  expand  during  the  heat  of  sum- 
mer without  crowding  each  other  out  of 
place.  Wagon  tires  are  made  smaller  than  H  ^^U^l) 

the  wheels   which   they  are   to.  fit.    They 

J  J      FIG.  69.    Expansion 

are  then  heated  until  they  become  large  of 


enough  to  be  driven  on,   and  in   cooling 
they  shrink  again  and  thus  grip  the  wheels  with  immense 
force.    A  common  lecture-room  demonstration  of  expansion 
is  the  following: 

Let  the  ball  B,  which  when  cool  just  slips  tlfrough  the  ring  R,  be 
heated  in  a  Bunsen  flame.  It  will  now  be  found  too  large  to  pass 
through  the  ring;  but  if  the  ring  is  heated,  or  if  the  ball  is  again 
cooled,  it  will  pass  through  easily  (see  Fig.  69). 

v/ 

98.  Evaporation  of  solids,  —  sublimation.    That  the  mole- 

cules of  a  solid  substance  are  found  in  a  vaporous  condition 
above  the  surface  of  the  solid,  as  well  as  above  that  of  a  liquid, 
is  proved  by  the  often  observed  fact  that  ice  and  snow  evapo- 
rate even  though  they  are  kept  constantly  below  the  freezing 
point.  Thus  wet  clothes  dry  in  winter  after  freezing.  'An 
even  better  proof  is  the  fact  that  the  odor  of  camphor  can  be 
detected  many  feet  away  from  the  camphor  crystals.  The 


72  MOLECULAR  MOTIONS 

evaporation  of  solids  may  be  rendered  visible  by  the  following 
striking  experiment: 

Let  a  few  crystals  of  iodine  be  placed  on  a  watch  glass  and  heated 
gently  with  a  Bunsen  flame.  The  visible  vapor  of  iodine  will  appear 
above  the  crystals,  though  none  of  the  liquid  is  formed.  A  great  many 
substances  at  high  temperatures  pass  thus  from  the  solid  to  the  gaseous 
condition  without  passing  through  the  liquid  stage  at  all.  This  process 
is  called  sublimation. 

99.  Diffusion  of  solids.    It  has  recently  been  demonstrated 
that  if  a  layer  of  lead  is  placed  upon  a  layer  of  gold,  mole- 
cules of  gold  may  in  time  be  detected  throughout  the  whole 
mass  of  the  lead.    This  diffusion  of  solids  into  one  another  at 
ordinary  temperature   has   been   shown   only   for   these   two 
metals,  but  at  higher  temperatures,  for  example  500°  C.,  all  of 
the  metals,  show  the  same  characteristics  to  quite  a  surprising 
degree. 

The  evidence  for  the  existence  of  molecular  motions  in 
solids  is  then  no  less  strong  than  in  the  case  of  liquids. 

100.  The   three  states  of   matter.    Although  it  has  been 
shown  that  in  accordance  with  current  belief  the  molecules  of 
all  substances  are  in  very  rapid  motion,  and  that  the  tempera- 
ture of  a  given  substance,  whether  in  the  solid,  liquid,  or 
gaseous  condition,  is  determined  by  the  average  velocity  of 
its  molecules,  yet  differences  exist  in  the  kind  of  motion  which 
the  molecules  in  the  three  states  possess.    Thus  in  the  solid 
state  it  is  probable  that  the  molecules  oscillate  with  great 
rapidity  about  certain  fixed  points,  always  being  held  by  the 
attractions  of  their  neighbors,  that  is,  by  the  cohesive,  forces 
(see  §  139),  in  practically  the  same  positions  with  reference 
to  other  molecules  in  the  body.    In  rare  instances,  however,  as 
the  facts  of  diffusion  show,  a  molecule  breaks  away  from  its 
constraints.    In  liquids,  on  the  other  hand,  while  the  mole- 
cules are,  in  general,  as  close  together  as  in  solids,  they  slip 
about  with  perfect  ease  over  one  another  and  thus  have  no 


MOLECULAR  MOTIONS  IN  SOLIDS  73 

fixed  positions.  This  assumption  is  necessitated  by  the  fact 
that  liquids  adjust  themselves  readily  to  the  shape  of  the 
containing  vessel.  In  gases  the  molecules  are  comparatively 
far  apart,  as  is  evident  from  the  fact  that  a  cubic  centimeter 
of  water  occupies  about  1600  cubic  centimeters  when  it  is 
transformed  into  steam ;  and  furthermore,  they  exert  prac- 
tically no  cohesive  force  upon  one  another,  as  is  shown  by 
the  indefinite  expansibility  of  gases. 

QUESTIONS  AND  PROBLEMS 

1.  Does  dew  "fall"? 

2.  Why  does  sprinkling  the  street  on  a  hot  day  make  the  air  cooler  ? 

3.  Why  is  the  heat  so  oppressive  on  a  very  damp  day  in  summer  ? 

4.  Would  fanning  produce  any  feeling  of  coolness  if  there  were  no 
moisture  on  the  face? 

5.  If  there  were  moisture  on  the  face,  would  fanning  produce  any 
feeling  of  coolness  in  a  saturated  atmosphere  ? 

6.  If  a  glass  beaker  and  a  porous  earthenware  vessel  are  filled  with 
equal  amounts  of  water  at  the  same  temperature,  in  the  course  of  a  few 
minutes  a  noticeable  difference  of  temperature,  will  exist  between  the 
two  vessels.   Which  will  be  the  cooler,  and  why  ?  Will  the  difference  in 
temperature  between  the  two  vessels  be  greater  in  a  dry  or  in  a  moist 
atmosphere  ? 

7.  Why  are  icebergs  frequently  surrounded  with  fog? 

8.  What  weight  of  water  is  contained  in  a  rodm  5  x  5  x  3  m.  if  the 
relative  humidity  is  60%  and  the  temperature  20°  C.  ?  (See  table,  p.  62.) 

9.  Why  will  an  open,  narrow-necked  bottle  containing  ether  not 
show  as  low  a  temperature  as  an  open  shallow  dish  containing  the 
same  amount  of  ether? 

10.  A  morning  fog  generally  disappears  before  noon.    Explain  the 
reason  for  its  disappearance. 

11.  What  becomes  of  the  cloud  which  you  see  about  a  blowing  loco- 
motive whistle  ?    Is  it  steam  ? 

12.  Dew  will  not  usually  collect  on  a  pitcher  of  ice  water  standing 
in  a  warm  room  on  a  cold  winter  day.    Explain. 


CHAPTER  V 

FORCE  AND  MOTION 

DEFINITION  AND  MEASUREMENT  OF  FORCE 

101 .  Distinction  between  a  gram  of  mass  and  a  gram  of  force. 
If  a  gram  of  mass  is  held  in  the  outstretched  hand,  a  down- 
ward pull  upon  the  hand  is  felt.    If  the  mass  is  50,000  g.  in- 
stead of  1,  this  pull  is  so  great  that  the  hand  cannot  be  held 
in  place.    The  cause  of  this  pull  we,  assume  to  be  an  attractive 
force  which  the  earth  exerts  on  the  matter  held  in  the  hand, 
and  we  define  the  gram  of  force  as  the  amount  of  the  earth's  pull 
at  its  surface  upon  one  gram  of  mass. 

Unfortunately,  in  ordinary  conversation  we  often  fail  alto- 
gether to  distinguish  between  the  idea  of  mass  and  the  idea 
of  force,  and  use  the  same  word  "  gram  "  to  mean  sometimes 
a  certain  amount  of  matter,  and  at  other  times  the  pull  of  the 
earth  upon  this  amount  of  matter.  That  the  two  ideas  are,  how- 
ever, wholly  distinct  is  evident  from  the  consideration  that 
the  amount  of  matter  in  a  body  is  always  the  same,  no  matter 
where  the  body  is  in  the  universe,  while  the  pull  of  the  earth 
upon  that  amount  of  matter  decreases  as  we  recede  from  the 
earth's  surface.  It  will  help  to  avoid  confusion  if  we  reserve 
the  simple  term  "  gram  "  to  denote  exclusively  an  amount  of 
matter,  that  is,  a  mass,  and  use  the  full  expression  "  gram  of 
force"  wherever  we  have  in  mind  the  pull  of. the  earth  upon 
this  mass. 

102.  Method  of  measuring  forces.    When  we  wish  to  com- 
pare accurately  the  pulls  exerted  by  the  earth  upon  different 
masses,  we  find  such  sensations   as  those  described  in  the 

74 


DEFINITION  AND  MEASUREMENT  OF  FORCE      75 


preceding  paragraph  very  untrustworthy  guides.  An  accurate 
method,  however,  of  comparing  these  pulls  is  that  furnished 
by  the  stretch  produced  in  a  spiral  spring.  Thus  the  pull 
of  the  earth  upon  a  gram  of  mass  at  its  sur- 
face will  stretch  a  given  spring  a  given  dis- 
tance ab  (Fig.  70).  The  pull  of  the  earth 
upon  2  .grams  of  mass  is  found  to  stretch  the 
spring  a  larger  distance  ac,  upon  3  grams  a 
still  larger  distance  ad,  etc.  We  have  only 
to  place  a  fixed  surface  behind  the  pointer 
and  make  lines  upon  it  corresponding  to  the 
points  to  which  it  is  stretched  by  the  pull  of 

the  earth  upon  different  masses  in  order  to    FIG.  70  Method  of 

_.      _  .  N  measuring  forces 

graduate  a  spring  balance  (rig.  71),  so  that 

it  will  thenceforth  measure  the  values  of  any  pulls  exerted 
upon  it,  no  matter  how  these  pulls  may  arise.  Thus,  if  a  man 
stretch  the  spring  so  that  the  pointer  is  opposite  the  mark 
corresponding  to  the  pull  of  the  earth  upon  2  grains  of  mass, 
we  say  that  he  exerts  2  grams  of  force.  If  he 
stretch  it  the  distance  corresponding  to  the  pull 
of  the  earth  upon  3  grams  of  mass,  he  exerts 
3  grams  of  force,  etc.  The  spring  balance  thus 
becomes  an  instrument  for  measuring  forces. 

103.  The  gram  of  force  varies  slightly  in  differ- 
ent localities.   With  the  spring  balance  it  is  easy          ~y 
to  verify  the  statement  made  above,  that  the  force  / 

of  the  earth's  pull  decreases  as  we  recede  from 

the  earth's  surface ;  for  upon  a  high  mountain    FlG-  71-   The 

...  •     •    T      ^    spring  balance 

the  stretch  produced  by  a  given  mass  is  indeed 

found  to  be  slightly  less  than  at  the  sea  level.  Furthermore, 
if  the  balance  is  simply  carried  from  point  to  point  over  the 
earth's  surface,  the  stretch  is  still  found  to  vary  slightly.  For 
example,  at  Chicago  it  is  about  one  part  in  1000  less  than  it 
is  at  Paris,  and  near  the  equator  it  is  five  parts  in  1000  less 


76  FOBCE  AND  MOTION 

than  it  is  near  the  pole.  This  is  due  in  part  to  the  earth's 
rotation,  and  in  part  to  the  fact  that  the  earth  is  not  a  per- 
fect sphere,  and  in  going  from  the  equator  toward  the  pole 
we  are  coming  closer  and  closer  to  the  center  of  the  earth. 
We  see,  therefore,  that  the  gram  of  force  is  not  an  absolutely 

invariable  unit  of  force. 

H 

COMPOSITION  AND  RESOLUTION  OF  FORCES 

104.  Graphic  representation  of  force.    A  force  is  completely 
denned  when  its  magnitude,  its  direction,  and  the  point  at  which 
it  is  applied  are  given.   Since  the  three  characteristics  of  a 
straight  line  are  its  length,  its  direction,  and  the  point  at  ivhich 
it  starts,  it  is  obviously  possible  to  represent 

forces  by  means  of  straight  lines.  Thus,  if 
we  wish  to  represent  the  fact  that  a  force  FlG-  72-  Graphic 
of  8  pounds,  acting  in  an  easterly  direction,  is 
applied  at  the  point  A  (Fig.  72),  we  draw  a 
line  8  units  long,  beginning  at  the  point  A  and  extending  to 
the  right.  The  length  of  this  line  then  represents  the  magni- 
tude of  the  force ;  the  direction  of  the  line,  the  direction  of 
the  force;  and  the  starting  point  of  the  line,  the  point  at 
which  the  force  is  applied. 

105.  Resultant  of  two  forces  acting  in  the  same  line.    The 
resultant  of  two  forces  is  defined  as  that  single  force  which  will 
produce  the  same  effect  upon  a  body  as  is  produced  by  the  joint 
action  of  the  two  forces. 

If  two  spring  balances  are  attached  to  a  small  ring  and 
pulled  in  the  same  direction  until  one  registers  10  g.  of  force 
and  the  other  5,  it  will  be  found  that  a  third  spring  balance 
attached  to  the  same  point  and  pulled  in  the  opposite  direc- 
tion will  register  exactly  15  g.  when  there  is  equilibrium ; 
that  is,  resultant  of  two  similarly  directed  forces  is  equal  to  the 
sum  of  the  two  forces. 


COMPOSITION  AND  RESOLUTION  OF  FORCES     77 


Similarly,  the  resultant  of  two  oppositely  directed  forces  applied 
at  the  same  point  is  equal  to  the  difference  between  them,  and  its 
direction  is  that  of  the  greater  force. 

106  o  Equilibrant.  In  the  last  experiment  the  pull  in  the 
spring  balance  which  registered  15  g.  was  not  the  resultant 
of  the  5  g.  and  10  g.  forces ;  it  was  rather  a  force  equal  and 
opposite  to  that  resultant.  Such  a  force  is 
called  an  equilibrant.  The  equilibrant  of  a 
force  or  forces  is  that  single  force  which  will 
just  prevent  the  motion  which  the  given  forces 
tend  to  produce.  It  is  equal  and.  opposite  to 
the  resultant. 

107.  The  resultant  of  forces  acting  at  an 
angle.  If  a  body  at  A  is  pulled  toward  the 
east  with  a  force  of  10  Ib.  (represented  in 
Fig.  73  by  the  line  AC)  and  toward  the  north  with  a  force 
of  10  Ib.  (represented  in  the  figure  by  the  line  AB),  the 
effect  upon  the  motion  of  the  body  must,  of  course,  be  the  same 
as  though  some  single  force  acted  somewhere  between  AC 
and  AB.  If  the  body 
moves  under  the  ac- 
tion of  the  two  equal 
forces,  it  may  be  seen 
from  symmetry  that  ^^-•"" 


FIG.  73.   Direction 

of  resultant  of  two 

equal  forces  at  right 

angles 


FIG.  74.    Resultant  of  two  forces  at  an  angle  is 

represented  by  the  diagonal  of  the  parallelogram 

of  which  the  forces  are  sides 


it  must  move  along 
a  line  midway  be- 
tween AC  and  AB, 
that  is,  along  the  line 

AR.  This  line  therefore  indicates  the  direction  as  well  as  the 
point  of  application  of  the  resultant  of  the  forces  AC  and  AB. 
If  the  two  forces  are  not  equal,  then  the  resultant  will  lie 
Nearer  the  larger  force.  As  a  matter  of  fact,  the  following 
experiment  will  show  that  if  the  two  given  forces  are  represented 
in  direction  and  in  magnitude  by  the  lines  AB  and  AC  (Fig.  74), 


78 


FOKCE  AND  MOTION 


then  their  resultant  will  be  exactly  represented  both  in  direction 
and  in  magnitude  by  the  diagonal  AR  of  the  parallelogram  of 
which  AB  and  AC  are  sides. 

Let  the  rings  of  two  spring  balances  be  hung  over  nails  B  and  C  in 
the  rail  at  the  top  of  the  blackboard  (Fig.  75),  and  let  a  weight  W 
be  tied  near  the  middle  of  the  string  joining  the  hooks  of  the  two 
balances.  The  force  of  the  earth's  attraction  for  the  weight  W  is  then 
exactly  equal  and  opposite  to  the  resultant  of  the  two  forces  exerted  by 

the  spring  balances ;  that  is,  O  W  is  the      R G      {j  ^ 

equilibrant  of  the  forces  exerted  by  the  bal- 
ances. Let  the  lines  OA  and  OD  be  drawn 
upon  the  blackboard  behind  the  string, 
and  upon  these  lines  let  distances  Oa  and 
Ob  be  laid  off  which  contain  as  many 
units  of  length  as  there  are  units  of  force 
indicated  by  the  balances  E  and  F  respec- 
tively. Then  let  a  parallelogram  be  con- 
structed upon  Oa  and  Ob  as  sides.  The 
diagonal  of  this  parallelogram  will  be 
found  in  the  first  place  to  be  exactly  ver- 
tical, that  is,  in  the  direction  of  the  re- 
sultant, since  it  is  exactly  opposite  to 


FIG.  75.    Experimental  proof 
of  parallelogram  law 


and  in  the  second  place,  the  length  of  the  diagonal  will  be  found 
to  contain  as  many  units  of  length  as  there  are  units  of  force  in  the 
earth's  attraction  for  W  (  W  must,  of  course,  be  expressed  in  the  same 
units  as  the  balance  readings).  Therefore  the  diagonal  OR  represents 
in  direction,  in  magnitude,  and  in  point  of  application  the  resultant  of 
the  two  forces  represented  by  Oa  and  Ob. 

108.  Component  of  a  force.  Whenever  a  force  acts  upon  a 
body  in  some  direction  other  than  that  in  which  the  body  is 
free  to  move,  it  is  clear  that  the  full  effect  of  the  force  can- 
not be  spent  in  producing  motion.  For  example,  suppose  that 
a  force  is  applied  in  the  direction  OR  (Fig.  76)  to  a  car  on  an 
elevated  track.  Evidently  OR  produces  two  distinct  effects 
upon  the  car:  on  the  one  hand,  it  moves  the  car  along  the1 
track ;  and  on  the  other,  it  presses  it  down  against  the  rails. 
These  two  effects  might  be  produced  just  as  well  by  two 


R 


COMPOSITION  AND  RESOLUTION  OF  FORCES     79 

separate  forces  acting  in  the  directions  OA  and  OB  respectively. 
The  value  of  the  single  force  which,  acting  in  the  direction  OA, 
will  produce  the  same  motion  of  the 
car  on  the  track  as  is  produced  by  OR, 
is  called  the  component  of  OR  in  the 
direction  OA.  Similarly,  the  value  of 
the  single  force  which,  acting  in  the  B 

direction  OB,  will  produce  the  same    FlG-  76-  Component  of  a 
pressure  against  the  rails  as  is  pro- 
duced by  the  force  OR,  is  called  the  component  of  OR  in  the 
direction  OB.    In  a  word,  the  component  of  a  force  in  a  given 
direction  is  the  effective  value  of  the  force  in  that  direction. 

109.  Magnitude  of  the  component  of  a  force  in  a  given  direc- 
tion. Since,  from  the  definition  of  component  just  given,  the 
two  forces,  one  to  be  applied  in  the  direction  OA  and  the  other 
in  the  direction  OB,  are  together  to  be  exactly  equivalent  to 
OR  in  their  effect  on  the  car,  their  magnitudes  must  be  repre- 
sented by  the  sides  of  a  parallelogram  of  which  OR  is  the 
diagonal.  For  in  §  107  it  was  shown  that  if  any  one  force  is 
to  t  have  the  same  effect  upon  a  body  as  two  forces  acting 
simultaneously,  it  must  be  represented  by  the  diagonal  of  a 
parallelogram  the  sides  of  which  represent  the  two  forces. 
Hence,  conversely,  if  two  forces  are  to  be 'equivalent  in  their 
joint  effect  to  a  single  force,  they  must  be  sides  of  the  paral- 
lelogram of  which  the  single  force  is  the  diagonal.  Hence  the 
following  rule :  To  find  the  component  of  a  force  in  any  given 
direction,  construct  upon  the  given  force  as  a  diagonal  a  rectangle 
the  sides  of  which  are  respectively  parallel  and  perpendicular  to 
the  direction  of  the  required  component.  The  length  of  the  side 
which  is  parallel  to  the  given  direction  represents  the  magnitude 
of  the  component  which  is  sought.  Thus,  in  the  above  illustra- 
tion, the  line  Om  completely  represents  the  component  of  OR 
in  the  direction  OA,  and  the  line  On  represents  the  component 
of  OR  in  the  direction  OB. 


80 


FORCE  AND  MOTION 


Again,  when  a  boy  pulls  on  a  sled  with  a  force  of  10  Ib. 
in  the  direction  OE  (Fig.  77),  the  force  with  which  the 
sled  is  urged  forward  is  represented  by  the  length  of  Om, 
which  is  seen  to  be 
but  9.3  Ib.  instead  of 
10  Ib. 

To  apply  the  test  of 
experiment  to  the  con- 
clusions of  the  preceding 
paragraph,  let  a  wagon 
be  placed  upon  an  in- 


FIG.  77.    Horizontal  component  of  pull  on  a  sled 


clined  plane  (Fig.  78),  the  height  of  which,  be,  is  equal  to  one  half  its 
length  ab.  In  this  case  the  force  acting  on  the  wagon  is  the  weight 
of  the  wagon,  and  its  direction  is  downward.  Let  this  force  be  repre- 
sented by  the  line  OR. .  Then  by  the  construction  of  the  preceding 
paragraph,  the  line  Om  will  represent  the  value  of  the  force  which  is 
pulling  the  carriage  down  the  plane,  and  the  line  On  the  value  of 
the  force  which  is  producing  pressure  against  the  plane.  Now  since 
the  triangle  ROm  is  similar  to  the  triangle  abc  (for  /.mOR  =  Zabc, 
Z.  RmO  —  Z.acb,  and  Z.  ORm  =  Z.  &ac),  we  have 

Om  _  be 
OR~'ab' 

that  is,  in  this  case,  since  be  is  equal  to  one  half  of  ab,  Om  is  one  half 
of  OR.  Therefore  the  force  which  is  necessary  to  prevent  the  wagon 
from  sliding  down  the  plane  should  be  equal 
to  one  half  its  weight.  To  test  this  con- 
clusion let  the  wagon  be  weighed  on  the 
spring  balance  and  then  placed  on  the  plane 
in  the  manTler  shown  in  the  figure.  The 
pull  indicated  by  the  balance  will,  indeed, 
be  found  to  be  one  half  of  the  weight  of 
the  wagon. 

The  equation  Om  /OR  =  bc/ab  gives  us  the 
following  rule  for  finding  the  force  necessary 
to  prevent  a  body  from  moving  down  an  in- 


FIG.  78.    Component   of 
weight  parallel  to  an  in- 
clined plane 


clined  plane,  namely,  the  force  which  must  be  applied  to  a  body  to  hold  it  in 
place  upon  an  inclined  plane  bears  the  same  ratio  to  the  weight  of  the  body 
that  the  height  of  the  plane  bears  to  its  length. 


COMPOSITION  AND  BESOLUTION  OF  FOECES     81 


110.  Component  of  gravity  effective  in  producing  the  motion 
of  the  pendulum.  When  a  pendulum  is  drawn  aside  from  its 
position  of  rest  (Fig.  79),  the  force  acting  on  the  bob  is  its 
weight,  and  the  direction  of  this  force  is  vertical. 
Let  it  be  represented  by  the  line  OR.  The  com- 
ponent of  this  force  in  the  direction  in  which  the 
bob  is  free  to  move  is  On,  and  the  component  at 
right  angles  to  this  direction  is  Om.  The  second 
component  Om  simply  produces  stretch  in  the' 
string  and  pressure  upon  the  point  of  suspen- 
sion. The  first  component  On  is  alone  responsi- 
ble for  the  motion  of  the  bob.  A  consideration 
of  the  figure  shows  that  this  component  becomes 
larger  and  larger  the  greater  the  displacement 
of  the  bob.  When  the  bob  is  directly  beneath 
the  point  of  support  the  component  producing 
motion  is  zero.  Hence  a  pendulum  can  be  permanently  at  rest 
only  when  its  bob  is  directly  beneath  the  point  of  suspension.* 


FIG.  79.  Force 
acting  on  dis- 
placed pendu- 
lum 


QUESTIONS  AND  PROBLEMS 

1.  In  Fig.  80  the  line  on  represents  the  pull  of  gravity  on  a  kite, 
and  the  line  om  represents  the  pull  of  the  boy  on  the  string.    What  is 
the  name  given  to  the  ^ 
force  represented  by                                                                           ,-'**"\ 

the  line  oR  ? 

2.  If  the  force  of 
the  wind  against  the 
kite    is    represented 
by  the  line  AB,  and 
it    is   considered    to 


\B 


FlG>  go>   Forceg  acting  on  a  kita 


be  applied  at  o,  what 
must  be  the  relation 
between  the  force  oR  and  the  component  of  AB  parallel  to  oR  when 
the  kite  is  in  equilibrium  under  the  action  of  the  existing  forces. 
3.  If  the  wind  increases,  why  does  the  kite  rise  higher? 

*  It  is  recommended  that  the  study  of  the  laws  of  tlie  pendulum  be  introduced 
into  the  laboratory  work  at  about  this  point  (see  Experiment  12,  authors'  manual). 


82 


FORCE  AND  MOTION 


4.  Represent  graphically  a  force  of  30  Ib.  acting  southeast  and  a 
force  of  40  Ib.  acting  southwest  at  the  same  point.    What  will  be  the 
magnitude  of  the  resultant,  and  what  will  be  its  approximate  direction  ? 

5.  The  engines  of  a  steamer  can  drive  it  12  mi.  an  hour.    How  fast 
can  it  go  up  a  stream  in  which  the  current  is  5  ft.  per  second  ?    How 
fast   can  it  come  down   the 

same  stream? 

6.  The    wind     drives    a 
steamer    east   with    a    force 
which  would  carry  it  12  mi. 
per  hour,  and  its  propeller  is 
driving  it  south  with  a  force 
which  would  carry  it  15  mi. 
per  hour.    What  distance  will 
it  actually  travel  in  an  hour  ? 

Draw  a  diagram  to  represent  FlG  81  Force  necessary  to  prevent  a  bar- 
the  exact  path.  rel  f  rom  rolling  down  an  inclined  plane 

7.  A  boy  pulls  a  loaded 

sled  weighing  200  Ib.  up  a  hill  which  rises  1  ft.  in  5  measured  along  the 
slope.  Neglecting  friction,  how  much  force  must  he  exert  ? 

8.  If  the  barrel  of  Fig.  81  weighs  200  Ib.,  with  what  force  must 
a    man   push    parallel   to    the    skid    to    keep   the    barrel   in   place    if 
the   skid  is   9  ft.  long   and  the  n   R 
platform    3  ft.  high  ?                                              Direction  of  Flight 

9.  Why    does    a    workman 
lower  the  handles  of  a  wheel- 
barrow when  he  wishes  to  push 
the  load  over  an  obstacle? 

10.  Could   a   kite   be   flown 
from  an  automobile  when  there 
is  no  wind  ?   Explain. 

11.  Show  from  Fig.  82  what  in  fl.  ht 
force  supports  an  aeroplane  in 

flight.  (Remember  that  oR,  the  component  of  the  wind  pressure  AB 
perpendicular  to  the  plane,  is  the  only  force  acting  out  of  which  a 
support  for  the  aeroplane  can  be  derived.) 

12.  What  force  will  be  required  to  support  a  50-lb.  ball  on  an  inclined 
plane  of  which  the  length  is  10  times  the  height  ? 

13.  A  boy  is  able  to  exert  a  force  of  75  Ib.    Neglecting  friction,  how 
long  an  inclined  plane  must  he  have  in  order  to  push  a  truck  weighing 
350  Ib.  up  to  a  doorway  3  ft.  above  the  ground  ? 


A'  B' 

FIG.  82.    Forces  acting  on  an  aeroplane 


GRAVITATION  83 

GRAVITATION 

111.  Newton's  law  of  universal  gravitation.    In  order  to 
account  for  the  fact  that  the  earth  pulls  bodie9\toward  itself, 
and  at  the  same  time  to  account  for  the  fact  that  the  moon  and 
planets  are  held  in  their  respective  orbits  about  the  earth  and 
the  sun,  Sir  Isaac  Newton  (1642-1727)  first  announced  the 
law  which  is  now  known  as  the  law  of  universal  gravitation. 
This  law  asserts  first  that  every  body  in  the  universe  attracts  every 
other  body  with  a  force  which  varies  inversely  as  the  square  of  the 
distance  between  the  two  bodies.   This  means  that  if  the  distance 
between  the  two  bodies  considered  is  doubled,  the  force  will 
become  only  one  fourth  as  great ;  if  the  distance  is  made  three, 
four,  or  five  times  as  great,  the  force  will  be  reduced  to  one  ninth, 
one  sixteenth,  or  one  twenty-fifth  off  its  original  value,  etc. 

The  law  further  asserts  that  if  the  distance  between  two 
bodies  remains  the  same,  the  force  with  which  one  body  attracts 
the  other  is  proportional  to  the  product  of  the  masses  of  the  two 
bodies.  Thus  we  know  that  the  earth  attracts  3  cubic  centi- 
meters of  water  with  three  times  as  much  force  as  it  attracts 
1,  that  is,  with  a  force  of  3  grams.  We  know  also,  from  the 
facts  of  astronomy,  that  if  the  mass  of  the  earth  were  doubled, 
its  diameter  remaining  the  same,  it  would  attract  3  cubic  cen- 
timeters of  water  with  twice  as  much  force  as  it  does  at 
present,  that  is,  with  a  force  of  6  grams  (multiplying  the  mass 
of  one  of  the  attracting  bodies  by  3  and  that  of  the  other  by 
2  multiplies  the  forces  of  attraction  by  3  x  2,  or  6).  In  brief, 
then,  Newton's  law  of  universal  gravitation  is  as  follows :  Any 
two  bodies  in  the  universe  attract  each  other  with  a  force  which 
is  directly  proportional  to  the  product  of  the  masses  and  inversely 
proportional  to  the  square  of  the  distance  between  them. 

112.  Variation  of  the  force  of  gravity  with  distance  above  the 
earth's  surface.    If  a  body  is  spherical  in  shape  and  of  uniform 
density,  it  attracts  external  bodies  with  the  same  force  as 


84  FOKCE  AND  MOTION 

though  its  mass  were  concentrated  at  its  center.  Since,  there- 
fore, the  distance  from  the  surface  to  the  center  of  the  earth 
is  about  4000  miles,  we  learn  from  Newton's  law  that  the 
earth's  pull  upon  a  body  4000  miles  above  its  surface  is  but 
one  fourth  as  much  as  it  would  be  at  the  surface. 

It  will  be  seen,  then,  that  if  a  body  be  raised  but  a  few  feet 
or  even  a  few  miles  above  the  earth's  surface,  the  decrease  in 
its  weight  must  be  a  very  small  quantity,  for  the  reason  that 
a  few  feet  or  a  few  miles  is  a  small  distance  compared  with 
4000  miles.  As  a  matter  of  fact,  at  the  top  of  a  mountain 
4  miles  high  1000  grams  of  mass  is  attracted  by  the  earth 
with  998  grams  instead  of  1000  grams  of  force. 

113.  Center  of  gravity.    From  the  law  of  universal  gravita- 
tion it  follows  that  every  particle  of  a  body  upon  the  earth's 
surface  is  pulled  toward  the  earth.    It  is 
evident  that  the  sum  of  all  these  little  pulls 
on  the  particles  of  which  the  body  is  com- 
posed must  be  equal  to  the  total  pull  of  the 
earth  upon  the  body.   Now  it  is  always  pos- 
sible to  find  one  single  point  in  a  body  at 
which  a  single  force  equal  in  magnitude  to  yF 

the  weight  of  the  body  and  directed  upward  FIG.  83.  Center  of 
can  be  applied  so  that  the  body  will  remain  gravity  of  an  irreg- 
at  rest  in  whatever  position  it  is  placed. 
This  point  is  called  the  center  of  gravity  of  the  body.  Since 
this  force  counteracts  entirely  the  earth's  pull  upon  the  body, 
it  must  be  equal  and  opposite  to  the  resultant  of  all  the  small 
forces  which  gravity  is  exerting  upon  the  different  particles  of 
the  body.  Hence  the  center  of  gravity  may  be  defined  as  the 
point  of  application  of  the  resultant  of  all  the  little  downward 
forces ;  that  is,  it  is  the  point  at  which  the  entire  iveight  of  the 
body  may  be  considered  as  concentrated.  The  earth's  attraction 
for  a  body  is  therefore  always  considered  not  as  a  multitude  of 
little  forces  but  as  one  single  force  F  (Fig.  83)  equal  to  the 


GRAVITATION 


85 


pull  of  gravity  upon  the  body  and  applied  at  its  center  of 
gravity  G.  It  is  evident,  then,  that  under  the  influence  of  the 
earths  pull,  every  body  tends  to  assume  the  position  in  which  its 
center  of  gravity  is  as  low  as  possible. 

114.  Method  of  finding  center  of  gravity  experimentally. 
From  the  above  definition  it  will  be  seen  that  the  most  direct 
way  of  finding  the  center  of  gravity  of  any  flat  body,  like  that 
shown  in  Fig.  84,  is  to  find  the  point  upon  which  it  will  balance. 

Let  an  irregular  sheet  of  zinc  be  thus  balanced  on  the  point  of  a 
pencil  or  the  head  of  a  pin.  Let  a  small  hole  be  punched  through 
the  zinc  at  the  point  of  balance, 
and  let  a  needle  be  thrust  through 
this  hole.  When  the  needle  is  held 
horizontally  the  zinc  will  be  found 
to  remain  at  rest,  no  matter  in 
what  position  it  is  turned. 

To  illustrate  another  method 
for  finding  the  center  of  gravity 
of  the  zinc,  let  it  be  supported 
from  a  pin  stuck  through  a  hole 
near  its  edge,  that  is,  b  (Fig.  84). 
Let  a  plumb  line  be  hung  from 
the  pin,  and  let  a  line  bn  be  drawn  through  b  on  the  surface  of  the  zinc 
parallel  to  and  directly  behind  the  plumb  line.  Let  the  zinc  be  hung 
from  another  point  a,  and  another  line  am  be  drawn  in  a  similar  way. 

Since  the  earth's  attraction  may  be  considered  as  a  single 
force  applied  at  the  center  of  gravity,  the  zinc  can  remain  at 
rest  only  when  the  center  of  gravity  is  directly  beneath  the 
point  of  support  (see  §  113).  It  must  therefore  lie  somewhere 
on  the  line  am.  For  the  same  reason  it  must  lie  on  the  line 
bn.  But  the  only  point  which  lies  on  both  of  these  lines  is 
their  point  of  intersection  G.  Tlie  point  of  intersection,  then, 
of  any  two  vertical  lines  dropped  through  two  different  points  of 
suspension  locates  the  center  of  gravity  of  a  body. 

115.  Stable  equilibrium.    A  body  is  said  to  be  in  stable  equi- 
librium if  it  tends  to  return  to  its  original  position  when  very 


FIG.  84.   Locating  center  of  gravity 


88 


FOKCE  AND  MOTION 


7.  Explain  why  the  toy  shown  in  Fig.  89  will  not  lie  upon  its  side, 
but  instead  rises  to  the  vertical  position.    Does  the  center  of  gravity 
actually  rise? 

8.  What  purpose  is  served  by  the  tail  of  a  kite? 

9.  If  a  lead  pencil  is  balanced  on  its  point  on 
the  finger,  it  will  be  in  unstable  'equilibrium,  but 
if  two  knives   are  stuck  into  it,    as   in   Fig.  90,  it 
will  be  in  stable  equilibrium. 

Why? 

10.  Why  does  a  man  lean 
forward   when   he   climbs   a 
hill? 

1 1 .  Do  you  get  more  sugar 
to  the  pound  in  Calcutta  than 

in  Aberdeen?    Explain.  FIG.  89  Fi<;.  (.K) 


\ 


FALLING  BODIES 

118.  Galileo's  early  experiments.  Many  of  the  familiar  and 
important  experiences  of  our  lives  have  to  do  with  falling 
bodies.  Yet  when  we  ask  ourselves  the  simplest  question 
.which  involves  quantitative  knowledge  about  gravity,  such 
as,  for  example,  Would  a  stone  and  a 
piece  of  lead  dropped  from  the  same 
point  reach  the  ground  at  the  same  or 
at  different  times  ?  most  of  us  are  uncer- 
tain as  to  the  answer.  In  fact,  it  was  the 
asking  and  the  answering  of  this  very 
question  by  Galileo  about  1590  which 
may  be  considered  as  the  starting  point 
of  modern  science'. 

Ordinary  observation  teaches  that  light 
bodies  like  feathers  fall  slowly  and  heavy 
bodies  like  stones  fall  rapidly,  and  up    FIG.  91.  Leaning  tower 
to  Galileo's  time  it  was  taught  in  the    of  Pisa,  from  which  were 

.,?.„       ,      .       performed  some  of  Gali- 

Schools    that    bodies    fall    With       veloci-     ieo's  famous  experiments 
ties  proportional  to  their  weights."    Not          on  falling  bodies 


GALILEO  (1564-1642) 

Great  Italian  physicist,  astronomer,  and  mathematician;  "founder  of  experi- 
mental science";  was  son  of  an  impoverished  nobleman  of  Pisa;  studied  medi- 
cine in  early  youth,  but  forsook  it  for  mathematics  and  science ;  was  professor 
of  mathematics  at  Pisa  and  at  Padua ;  discovered  the  laws  of  falling  bodies  and 
the  laws  of  the  pendulum ;  was  the  creator  of  the  science  of  dynamics ;  constructed 
the  first  thermometer;  first  used  the  telescope  for  astronomical  observations; 
discovered  Jupiter's  satellites  and  the  spots  on  the  sun.  Modern  physics  begins 

with  Galileo 


FALLING  BODIES 


89 


content  with  book  knowledge,  however,  Galileo  tried  it 
himself.  In-  the  presence  of  the  professors  and  students  of 
the  University  of  Pisa  he  dropped  balls  of  different  sizes 
and  materials  from  the  top  of  the  tower  of  Pisa  (Fig.  91), 
180  feet  high,  and  found  that  they  fell  in  practically  the 
same  time.  He  showed  that  even  very  light  bodies  like  paper 
fell  with  velocities  which  approached  more 
and  more  nearly  those  of  heavy  bodies  the 
more  compactly  they  were  wadded  together. 
From  these  experiments  he  inferred  that  all 
bodies,  even  the  lightest,  would  fall  at  the 
same  rate  were  it  not  for  the  resistance  of 
the  air. 

That  the  air  resistance  is  indeed  the  chief  factor 
in  the  slowness  of  fall  of  feathers  and  other  light 
objects  can  be  shown  by  pumping  the  air  out  of  a 
tube  containing  a  fe'ather  and  a  coin  (Fig.  92).  The 
more  complete  the  exhaustion  the  more  nearly  do 
the  feather  and  coin  fall  side  by  side  when  the  tube  <[ 

is  inverted.   The  air  pump,  however,  was  not  invented 
until  sixty  years  after  Galileo's  time.  FlG   92     Feather 

- ,  ft     ^  TTT       and  coin  fall  to- 

119.  Exact  proof  of  Galileo's  conclusion.  We    getherinavacuum 
can  demonstrate  the  correctness  of  Galileo's 
conclusion  in  still  another,  way,  one  which  he  himself  used. 

Let  balls  of  iron  and  wood,  for  example,  be  started  together  down 
the  inclined  plane  of  Fig.  93.  They  will  be  found  to  keep  together 
all  the  way  down.  (If  they  roll  in  a  groove,  they  should  have  the 
same  diameter ;  otherwise,  size  is  immaterial.)  The  experiment  differs 
from  that  of  the  freely  falling  bodies  only  in  that  the  resistance  of 
the  air  is  here  more  nearly  negligible  because  the  balls  are  moving 
.more  slowly.  In  order  to  make  them  move  still  more  slowly  and  at 
the  same  time  to  eliminate  completely  all  possible  effects  due  to  the 
friction  of  the  plane,  let  us  follow  Galileo  and  suspend  the  different 
balls  as  the  bobs  of  pendulums  of  exactly  the  same  length,  two  meters 
long  at  least,  and  start  them"  swinging  through  equal  arcs.  Since  now 
the  bobs,  as  they  pass  through  any  given  position,  are  merely  moving 


90  FOKCE  AND  MOTION 

very  slowly  down  identical  inclined  planes  (Fig.  79),  it  is  clear  that 
this  is  only  a  refinement  of  the  last  experiment.  We  shall  find  that 
the  times  of  fall,  that  is,  the  periods,  of  the  pendulums  are  exactly 
the  same. 

We  conclude  from  the  above  experiment,  with  Galileo 
and  with  Newton  who  performed  it  with  the  utmost  care 
a  hundred  years  later,  that  in  a  vacuum  the  velocity  acquired 
per  second  by  a  freely  falling  body  is  exactly  the  same  for 
all  bodies. 

120.  Relation  between  distance  and  time  of  fall.  Having 
found  that,  barring  air  resistance,  all  bodies  fall  in  exactly  the 
same  way,  we  shall  next  try  to  find  what  relation  exists  be- 
tween distance  and  time  of  fall ;  and  since  a  freely  falling  body 
falls  so  rapidly  as  to  make  direct  measurements  upon  it  diffi- 
cult, we  shall  adopt  Galileo's  plan  of  studying  the  laws  of  fall- 
ing bodies  through  observing  the  motions  of  a  ball  rolling 
down  an  inclined  plane. 

Let  a  grooved  board  17  or  18  ft.  long  be  supported  as  in  Fig.  93,  one 
end  being  about  a  foot  above  the  other.  Let  the  side  of  the  board  be 
divided  into  feet,  and  let  the  block  B  be  set  just  16  ft.  from  the  start- 
ing point  of  the  ball  A.  Let  a  metronome  or  a  clock  beating  seconds  be 
started,  and  the  marble  released  at  the  instant  of  one  click  of  the  metro- 
nome. If  the  marble  does  not  hit  the  block  so  that  the  click  produced  by 
the  impact  of  the  ball  coincides  exactly  with  the  fifth  click  of  the  metro- 
nome, alter  the  inclination  until  this  is  the  case.  (This  adjustment  may 
well  be  made  by  the  teacher  before  class.)  Now  start  the  marble  again 
at  some  click  of  the  metronome  and  note  that  it  crosses  the  1-ft.  mark 
exactly  at  the  end  of  the  first  second,  the  4-ft.  mark  at  the  end  of  the 
second  second,  the  9-ft.  mark  at  the  end  of  the  third  second,  and  hits 
B  at  the  16-ft.  mark  at  the  end  of  the  fourth  second.  This  can  be  tested 
more  accurately  by  placing  B  successively  at  the  9-ft.,  the  4-ft.,  and  the 
1-ft.  mark,  and  noting  that  the  click  produced  by  the  impact  coincides 
exactly  with  the  proper  click  of  the  metronome. 

We  conclude  then,  with  Galileo,  that  the  distance  traversed 
by  a  falling  body  in  any  number  of  seconds  is  the  distance 
traversed  the  first  second  times  the  square  of  the  number  of 


FALLING  BODIES 


91 


seconds  ;  that  is,  if  D  represents  the  distance  traversed  the  first 
second,  S  the  total  space,  and  t  the  number  of  seconds,  S  =  Dt2. 
121.  Relation  between  velocity  and  time  of  fall.  In  the  last 
paragraph  we  investigated  the  distances  traversed  in  one, 
two,  three,  etc.  seconds.  Let  us  now  investigate  the  velocities 
acquired  on  the  same  inclined  plane  in  one,  two,  three,  etc., 
seconds. 

Let  a  second  grooved  board  M  be  placed  at  the  bottom  of  the  incline, 
in  the  manner  shown  in  Fig.  93.  To  eliminate  friction  it  should  be 
given  a  slight  slant,  just  sufficient  to  cause  the  ball  to  roll  along  it  with 


FIG.  93.    Spaces  traversed  and  velocities  acquired  by  falling  bodies  in  one, 
two,  three,  etc.  secpnds 

uniform  velocity.  Let  the  ball  be  started  at  a  distance  D  up  the  incline, 
D  being  the  distance  which  in  the  last  experiment  it  was  found  to  roll 
during  the  first  second.  It  will  then  just  reach  the  bottom  of  the  incline 
at  the  instant  of  the  second  click.  Here  it  will  be  freed  from  the  influ- 
ence of  gravity,  and  will  therefore  move  along  the  lower  Board  with  the 
velocity  which  it  had  at  the  end  of  the  first  second.  It  will  be  found 
that  when  the  block  is  placed  at  a  distance  exactly  equal  to  2  D  from 
the  bottom  of  the  incline,  the  ball  will  hit  it  at  the  exact  instant  of  the 
third  click  of  the  metronome,  that  is,  exactly  two  seconds  after  starting  ; 
hence  the  velocity  acquired  in  one  second  is  2  Z).  If  the  ball  is  started  at 
a  distance  4  Z)  up  the  incline,  it  will  take  it  two  seconds  to  reach  the 
bottom,  and  it  will  roll  a  distance  4  Z)  in  the  next  second ;  that  is,  in 
two  seconds  it  acquires  a  velocity  4  D.  In  three  seconds  it  will  be  found 
to  acquire  a  velocity  6  Z>,  etc. 

The  experiment  shows,  first,  that  the  gain  in  velocity  each 
second  is  the  same ;  and  second,  that  the  amount  of  this  gain 
is  numerically  equal  to  twice  the  distance  traversed  the  first 
second.  Motion,  like  the  above,  in  which  velocity  is  gained  at 
a  constant  rate,  is  called  uniformly  accelerated  motion. 


92 


FORCE  AND  MOTION 


In  uniformly  accelerated  motion  the  gain  each  second  in  the 
velocity  is  called 'the  acceleration.  It  is  numerically  equal  to 
twice  the  distance  traversed  the  first  second.  It  is  usually 
denoted  by  the  letter  a. 

122.  Formal  statement  of  the  laws  of  falling  bodies.  Put- 
ting together  the  results  of  the  last  two  paragraphs,  we  obtain 
the  following  table  in  which  D  represents  the  distance  trav,- 
ersed  the  first  second  in  any  uniformly  accelerated  motion/ 


NUMBER  OF 
SECONDS  (t) 

VELOCITY  AT  THE 
END  OF  EACH 
SECOND  (v) 

GAIN  IN  VELOCITY 

EACH  SECOND  (a) 

TOTAL  DISTANCE 
TRAVERSED  (S) 

1 

2  D 

2D 

ID 

2 

4D 

2D 

4D 

3 

61} 

2Z> 

9ZJ. 

4 

8D 

21> 

16D 

.  .   . 

.  .  . 

.  .  . 

i  •  • 

t 

2tD 

21> 

PD 

Since  D  was  shown  in  §  121  to  be  equal  to  one  half  of  the  accel- 
eration a,  we  have  at  once,  by  substituting  l  a  for  D  in  the  last 
Jine  of  the  table,  v=at,  (1) 

S=$at*.  (2) 

These  formulas  are  simply  the  algebraic  statement  of  the  facts 
brought  out  by  our  experiments,  but  the  reasons  for  these  facts  may 
be  seen  as  follows  : 

Since  in  uniformly  accelerated  motion  the  acceleration  a  is  the 
velocity  in  centimeters  per  second  gained  each  second,  it  follows  at 
once  that  when  a  body  starts  from  rest,  the  velocity  which  it  has  at 
the  end  of  t  seconds  is  given  by  v  =  at.  This  is  formula  (1). 

To  obtain  formula  (2)  we  have  only  to  reflect  that  distance  traversed 
is  always  equal  to  the  average  velocity  multiplied  «by  the  time.  When 
the  initial  velocity  is  zero,  as  in  this  case,  and  the  final  velocity  is  at, 
average  velocity  =  (0  +  at)  -*-  2  =  \  at.  Hence 

S  =  1  at2. 
This  is  formula  (2). 

These  are  the  fundamental  formulas  of  uniformly  accelerated  motion, 
but  it  is  sometimes  convenient  to  obtain  the  final  velocity  v  directly  from 


FALLING  BODIES  93 

the  total  distance  of  fall  S,  or  vice  versa.    This  may  of  course  be  done 

by  simply  substituting  in  (2)  the  value  of  t  obtained  from  (1),  namely  -• 

This  gives  a 

4  v  =  VZaS.  (3) 

123.  Acceleration  of  a  freely  falling  body.  If  in  the  above 
experiment  the  slope  of  the  plane  be  made  steeper,  the  results 
will  obviously  be  precisely  the  same,  except  that  the  accelera- 
tion has  a  larger  value.  If  the  board  is  tilted  until  it  becomes 
vertical,  the  body  becomes  a  freely  falling  body.  In  this  case 
the  distance  traversed  the  first  second  is  found  to  be  490  centi- 
meters, or  16.08  feet.  Hence  the  acceleration  expressed  in 
centimeters  is  980,  in  feet  32.16.  This  acceleration  of  free 
fall,  called  the  acceleration  of  gravity,  is  usually  denoted  by  the 
letter  g.  For  freely  falling  bodies,  then,  the  three  formulas 
of  the  last  paragraph  become 


f 


(5) 

(6) 


To  illustrate  the  use  of  these  formulas,  suppose  we  wish  to  know 
with  what  velocity  a  body  will  hit  the  earth  if  it  falls  from  a  height 
of  200  meters  or  20,000  centimeters.  From  (6)  we  get 


v  =  V2  x  980  x  20,000  =  6261  cm.  per  sec. 

124.  Height  of  ascent.  If  we  wish  to  find  the  height  S  to  which  a  body 
projected  vertically  upward  will  rise,  we  reflect  that  the  time  of  ascent 
must  be  the  initial  velocity  divided  by  the  upward  velocity  which  the 

body  loses  per  second,  that  is,  t  =  - ;  and  the  height  reached  must  be  this 

9    f 

multiplied  by  the  average  velocity r- ;  that  is, 

S==irg>     or     y=V2^S.  (7) 

Since  (7)  is  the  same  as  (6), -we  learn  that  in  a  vacuum  the  speed  with 
which  a  body  must  be  projected  upward  to  rise  to  a  given  height  is  the 
same  as  the  speed  which  it  acquires  in  falling  from  the  same  height. 


94  FORCE  AND  MOTION 

125.  The  laws  of  the  pendulum.  The  first  law  of  the 
pendulum  was  found  in  §  119,  namely, 

1 .  The  periods  of  pendulums  of  equal  lengths  swinging  through 
short  arcs  are  independent  of  the  weight  and  material  of  the  bobs. 

Let  the  two  pendulums  of  §  119  be  set  swinging  through  arcs  of 
lengths  5  centimeters  and  25  centimeters  respectively.  We  shall  find 
thus  the  second  law  of  the  pendulum,  namely, 

2.  The  period  of  a  pendulum  swinging  through  a  short  arc  is 
independent  of  the  amplitude  of  the  arc. 

Let  pendulums  ^  and  J  as  long  as  the  above  be  swung  with  it.  The 
long  pendulum  will  be  found  to  make  only  one  vibration,  while  the  others 
are  making  two  and  three  respectively.  The  third  law  of  the  pendulum 
is  therefore 

3.  The  periods  of  pendulums  are  directly  proportional  to  tlu> 
square  roots  of  their  lengths. 

The  accurate  determination  of  g  is  never  made  by  direct  measure- 
ment, for  the  laws  of  the  pendulum  just  established  make  this  instru- 
ment by  far  the  most  accurate  one  obtainable  for  this  dB^mination. 
It  is  only  necessary  to  measure  the  length  of  a  long  pendu^M  and  the 
time  t  between  two  successive  passages  of  the  bob  across  -th^nnd-point 

and  then  to  substitute  in  the  formula  t  =  TT  ^l—  in  order  to  obtain  g  with  a 

high  degree  of  precision.  The  deduction  of  this  formula  is  not  suitable  for 
an  elementary  text,  but  the  formula  itself  may  well  be  used  for  checking 
the  above  value  of  g. 

V>  -  °f  *  QUESTIONS  AND  PROBLEMS 

1.  A  boy  dropped  a  stone  from  a  bridge  and  noticed  that  it  struck 
the  water  in  just  3  sec.    How  fast  was  it  going  when  it  struck  ?  How 
high  was  the  bridge  above  the  water  ? 

2.  How  high  is  a  balloon  from  which  a  stone  falls  t6  earth  in  10  sec.? 

3.  With  what  speed  does  a  bullet  strike  the  earth,  if  it  is  dropped 
from  the  Eiffel  Tower,  335  m.  high? 

4.  If  the  acceleration  of  a  marble  rolling  down  an  inclined  plane  is 
20  cm.  per  second,  what  velocity  will  it  have  at  the  bottom,  the  plane 
being  7  m.  long  ? 

5.  If  a  man  can  jump  3  ft.  high  on  the  earth,  how  high  could  he 
jump  on  the  moon,  where  g  is  ^  as  much  ? 


NEWTON'S  LAWS  OF  MOTION 


95 


6.  If  a  body  sliding  without  friction  down  an  inclined  plane  moves 
40  cm.  during  the  first  second  of  its  descent,  and  if  the  plane  is  500  cm. 
long  and  40.8  cin.  high,  what  is  the  value  of  #?    (Remember  that  the 
acceleration  down  the  incline  is   simply  the   com- 
ponent (§  108)  of  g  parallel  to  the  incline.) 

7.  How  far  will  a  body  fall  in  half  a  second? 

8.  Fig.  94  represents  the  pendulum  and  "escape- 
ment "  of  a  clock.   The  escapement  wheel  D  is  urged 
in  the  direction  of  the  arrow  by  the  clock  weights 
or  spring.    The  slight  pushes  communicated  by  the 
teeth  of  the  wheel  keep  the  pendulum  from  dying 
down.    Show  how  the  length  of  the  pendulum  con- 
trols the  rate  of  the  clock. 


v> 


NEWTON'S  LAWS  OF  MOTION 


126.  First  law  —  inertia.  It  is  a  matter  of 
everyday  observation  that  bodies  in  a  moving 
train  tend  t^move  toward  the  forward  end 
when  the^Bin  stops  and  totvard  the  rear 
end  wheia^^^train  starts ;  that  is,  bodies  in 
motion  svHP>  want  to  jkeep  on  moving,  and 
bodies  at  rest  to  remain  at  rest. 

Again,  a  block  will  go  farther  when  driven 
with  a  given  blow  along  a  surface  of  glare  }ce 
than  when  knocked  along  an  asphalt  pave-  -pIG  94 

ment.   The  reason  which  every  one  will  assign 
for  this  is  that  there  is  more  friction  between  the  block  and 
the  asphalt  than  between  the  block  and  the  ice.    But  when 
would  the  body  stop  if  there  were  no  friction  at  all  ? 

Astronomical  observations  furnish  the  most  convincing 
answer  to  this  question,  for  we  cannot  detect  any  retardation 
at  all  in  the  motions  of  the  planets  as  they  swing  around  the 
sun  through  empty  space. 

Furthermore,  since  mud  flies  off  tangentially  from  a  rotating 
carriage  wheel,  or  water  from  a  whirling  grindstone,  and  since, 
too,  we  have  to  lean  inward  to  prevent  ourselves  from  falling 


96  FOKCE  AND  MOTION 

outward  in  going  around  a  curve,  it  appears  that  bodies  in 
motion  tend  to  maintain  not  only  the  amount  but  also  the 
direction  of  their  motion. 

In  view  of  observations  of  this  sort  Sir  Isaac  Newton  in 
1686  formulated  the  following  statement  arid  called  it  the 
first  law  of  motion. 

Every  body  continues  in  its  state  of  rest  or  uniform  motion  in  a 
straight  line  unless  impelled  by  external  force  to  change  that  state. 

This  property,  tvhich  all  matter  possesses,  of  resisting  any  at- 
tempt to  start  it  if  at  rest,  to  stop  it  if  in  motion,  or  in  any  way  to 
change  either  the  direction  or  amount  of  its  motion,  is  called  inertia. 

127.  Centrifugal  force.    It  is  inertia  alone  which  prevents 
the  planets  from  falling  into  the  sun,  which  causes  a  rotating 
sling  to  pull  hard  on  the  hand  until  the  stone^^  released,  and 
which  then  causes  the  stone  to 

fly  off  tangeiitially.  It  is  iner- 
tia which  makes  rotating  liquids 
move  out  as  far  as  possible  from 
the  axis  of  rotation  (Fig.  95), 
which  makes  flywheels  some- 
times burst,  which  makes  the 

equatorial  diameter  of  the  earth 

FIG.  95.    Illustrating  centrifugal 
greater   than    the    polar,    which  force 

makes    the   heavier   milk   move 

out  farther  than  the  lighter  cream  in  the  dairy  separator,  etc. 
Inertia  manifesting  itself  in  this  tendency  of  the  parts  of  rotat- 
ing systems  to  move  away  from  the  center  of  rotation  is  called 
centrifugal  force. 

128.  Momentum.    The  quantity  of  motion  possessed  by,  a 
moving  body  is  defined  as  the  product  of  the  mass  and  the 
velocity  of  the  body.    It  is  commonly  called  momentum.    Thus 
a  10-gram  bullet  moving  50,000  centimeters  per  second  has 
500,000  units  of  momentum.    A  1000-kg.  pile  driver  moving 


SIR  ISAAC  NEWTON  (1642-1727) 

English  mathematician  and  physicist,  "  prince  of  philosophers  "  ; 
professor  of  mathematics  at  Cambridge  University ;  formulated 
the  law  of  gravitation;  discovered  the  binomial  theorem;  in- 
vented the  method  of  the  calculus ;  announced  the  three  laws  of 
motion  which  have  become  the  basis  of  the  science  of  mechanics ; 
made  important  discoveries  in  light ;  is  the  author  of  the  celebrated 
"  Principia  "  (Principles  of  Natural  Philosophy) ,  published  in  1687 


re 


NEWTON'S  LAWS  OF  MOTION  97 

1000  centimeters  per  second  has  1,000,000,000  units  of  mo- 
mentum, etc.  We  shall  always  express  momentum  in  C.G.S. 
units,  that  is,  as  a  product  of  grams  by  centimeters  per 
second. 

129.  Second  law.    Since  a  2-gram  mass  is  pulled  toward 
the  earth  with  twice  as  much  force  as  is  a  1-gram  mass,  and 
since  both,  when  allowed  to  fall,  acquire  the  same  velocity  in 
a  second,  it  follows  that  in  this  case  the  momentums  produced 
in  the  two  bodies  by  the  two  forces  are  exactly  proportional  to  the 
forces  themselves.    In  all  cases  in  which  forces  simply  overcome 
inertias  this  rule  is  found  to  hold.    Thus  a  3000-pound  pull 
on  an  automobile  on  a  level  road,  where  friction  may  be  neg- 
lected, imparts  in  a  second  just  twice  as  much  velocity  as 
does  a  1500-pound  pull.    In  view  of  this  relation  Newton's 
second  law  of  motion  was  stated  thus : 

Rate  of  change  of  momentum  is  propor- 
tional to  the  force  acting,  and  takes  place 
in  the  direction  in  which  the  force  acts. 

130.  The  third  law.    When  a  man 
jumps  from  a  boat  to  the  shore,  we     {' 


all  know  that   the    boat    experiences      '"c"     ^^p^^r     '"/>' 

a  backward  thrust ;  when  a  bullet  is 

shot  from  a  gun,  the  gun  recoils,  or      F'G-  ^^P^™*10*  °f 

"  kicks  "  ;  when  a  billiard  ball  strikes 

another,  it  loses  speed,  that  is,  is  pushed  back  while  the  second 

ball  is  pushed  forward.    The  following  experiment  will  show 

how  effects  of  this  sort  may  be  studied  quantitatively. 

Let  a  steel  ball  A  (Fig.  96)  be  allowed  to  fall  from  a  position  C 
against  another  exactly  similar  ball  B.  In  the  impact  A  will  lose  prac- 
tically all  of  its  velocity,  and  B  will  move  to  a  position  D,  which  is  at 
practically  the  same  height  as  C.  Hence  the  velocity  acquired  by  B  is 
almost  exactly  equal  to  that  which  A  had  before  impact.  These  veloc- 
ities would  be  exactly  equal  if  the  balls  were  perfectly  elastic.  It  is 
found  to  be  exactly  true  that  the  momentum  acquired  by  B  plus  that 
retained  by  A  is  exactly  equal  to  the  momentum  which  A  had  before 

t 


98  FOBCE  AND  MOTION 

the  impact.  The  momentum  acquired  by  B  is  therefore  exactly  equal 
to  that  lost  by  A.  Since  by  the  second  law  change  in  momentum  is  pro- 
portional to  the  fdrce  acting,  this  experiment  shows  that  A  pushed  for- 
ward on  B  with  exactly  the  same  force  with  which  B  pushed  back  on  A . 

Now  the  essence  of  Newton's  third  law  is  the  assertion  that 
in  the  case  of  the  man  jumping  from  the  boat  the  mass  of  the 
man  times  his  velocity  is  equal  to  the  mass  of  the  boat  times 
its  velocity,  and  that  in  the  case  of  the  bullet  and  gun  the 
mass  of  the  bullet  times  its  velocity  is  equal  to  the  mass  of 
the  gun  times  its  velocity.  The  truth  of  this  assertion  has 
been  established  by  a  great  variety  of  experiments  in  addition 
to  the  one  on  impact  given  above. 

Newton  stated  his  third  law  thus :  To  every  action  there  is 
an  equal  and  opposite  reaction. 

Since  force  is  measured  by  the  rate  at  which  momentum 
changes,  this  is  only  another  way  of  saying  that  whenever  a 
body  acquires  momentum  some  other  body  acquires  an  equal  and 
opposite  momentum. 

It  is  not  always  easy  to  see  at  first  that  setting  one  body 
into  motion  involves  imparting  an  equal  and  opposite  momen- 
tum to  another  body.  For  example,  when  a  gun  is  held  against 
the  earth  and  a  bullet  shot  upward,  we  are  conscious  only 
of  the  motion  of  the  bullet;  the  other  body  is  in  this  case 
the  earth,  and  its  momentum  is  the  same  as  that  of  the  bullet. 
On  account,  however,  of  the  greatness  of  the  earth's  mass  its 
velocity  is  infinitesimal. 

131.  The  dyne.  Since  the  gram  of  force  varies  somewhat  with  locality, 
it  has  been  found  convenient  for  scientific  purposes  to  take  the  second 
law  as  the  basis  for  the  definition  of  a  new  unit  of  force.  It  is  called  an 
absolute  or  C.G.S.  unit,  because  it  is  based  upon  the  fundamental  units 
of  length,  mass,  and  time,  and  is  therefore  independent  of  gravity.  It 
is  named  the  dyne,  and  is  defined  as  the  force  which,  acting  for  one  second 
upon  any  mass,  imparts  to  it  one  unit  of  momentum;  or  the  force  which,  acting 
for  one  second  upon  a  one-gram  mass,  produces  a  change  in  its  velocity  of 
one  centimeter  per  second. 


NEWTON'S  LAWS  OF   MOTION  99 

132.  A  gram  of  force  equivalent  to  980  dynes.    A  gram  of  force  was 
defined  as  the  pull  of  the  earth  upon  1  grain  of  mass.    Since  this  pull  is 
capable  of  imparting  to  this  mass  in  1  second  a  velocity  of  980  centi- 
meters per  second,  that  is,  980  units  of  momentum,  and  since  a  dyne 
is  the  force  required  to  impart  in  1  second '1*  unit  of  momentum,  it  is 
clear  that  the  gram  of  force  is  equivalent  to  980  dynes  of  force.    The 
dyne  is  therefore  a  very  small  unit,  about  equal  to  the  force  with  which 
the  earth  attracts  a  cubic  millimeter  of  water. 

133.  Algebraic  statement  of  the  second  law.    If  a  force  F  acts  for  t  sec- 
onds on  a  mass  of  m  grams,  and  in  so  doing  increases  its  velocity  v  cen- 
timeters per  second,  then,  since  the  total  momentum  imparted  in  a  time 

t  is  ?nv,  the  momentum  imparted  per  second  is  —  ;  and  since  force  in 
dynes  is  equal  to  momentum  imparted  per  second,  we  have 


But  since  -  is  the  velocity  gained  per  second,  it  is  equal  to  the  acceler- 
ation a.    Equation  (8)  may  therefore  be  written 

F  =  ma.  (9) 

This  is  merely  stating  in  the  form  of  an  equation  that  'force  is 
measured  by  rate  of  change  in  momentum.  Thus,  if  an  engine,  after  pull- 
ing for  30  sec.  on  a  train  having  a  mass  of  2,000,000  kg.,  has  given  it  a 
velocity  of  60  cm.  per  second,  the  force  of  the  pull  is  2,000,000,000  x  |^  = 
4,000,000,000  dynes.  To  reduce  this  force  to  grams  we  divide  by39°80, 
and  to  reduce  it  to  kilos  we  divide  further  by  1000.  Hence  the  pull 
exerted  by  the  engine  on  the  train  =  4>0S8o>touu°0^  =  4081  kg.,  or  4.081 
metric  tons. 

QUESTIONS  AND  PROBLEMS 

1.  Balance  a  calling  card  on  the  finger   and   place  a  coin  upon 
it.     Snap   out   the    card,    leaving    the    coin    balanced    on  the    finger. 
What  principle  is  illustrated  ? 

2.  Why  does  not  the  car  C 
of  Fig.  97  fall?   What  carries 
it  from  BtoD? 

3.  Why     does     a    flywheel 
cause  machinery  to  run  more 

steadily?  FJG   9?_   A  yery  ancient  loop  the  loops 

4.  What  principle  is  applied 

when  one  tightens  the  head  of  a  hammer  by  pounding  on  the  handle  ? 


100 


FOBCE  AND  MOTION 


FIG. 


5.  Is  it  any  easier  to  walk  toward  the  rear  than  toward  the  front 
of  a  rapidly  moving  train  ?    Why  ? 

6.  Why  does  pounding  a  carpet  free  it  from  dust? 

7.  Suspend  a  weight  by  a  string  (Fig.  98).    Attach  a  piece  of  the 
same  string  to  the  bottom  of  the  weight.    If  the  lower  string  is  pulled 
with  a  sudden  jerk,  it  breaks ;  but  if  the  pull  is  steady,  the 

upper  string  will  break.    Explain. 

8.  If  a  weight  is  dropped  from  the  roof  to  the  floor  of  a 
moving  car,  will  it  strike  the  point  on  the  floor  which  was 
directly  beneath  its  starting  point? 

9.  Why  is  a  running  track  banked  at  the  turns? 

10.  If  the  earth  were  to  cease  rotating,  would  bodies  on 
the  equator  weigh  more  or  less  than  now  ?   Why  ? 

1 1 .  How  is  the  third  law  involved  in  rotary  lawn  sprinklers  ? 

12.  The  modern  way  of  drying  clothes  is  to  place  them  in 

a  large  cylinder  with  holes  in  the  sides,  and  then  to  set  it  in  rapid 
rotation.    Explain. 

13.  If  one  ball  is  thrown  horizontally  from  the  top  of  a  tower  and 
another  dropped  at  the  same  instant,  which  will  strike  the  earth  first  ? 
(Remember  that  the  acceleration  produced  by  a  force  is  in  the  direction 
in  which  the  force  acts  and  proportional  to  it, 

whether  the  body  is  at  rest  or  in  motion.  See 
second  law.)  If  possible,  try  the  experiment 
with  an  arrangement  like  that  of  Fig.  99. 

14.  If  a  rifle  bullet  is  fired  horizontally  from 
a  tower  19.6  m.  high  with  a  speed  of  300  m., 
how  far  from  the  base  of  the  tower  would  it 
strike  the  earth  if  there  were  no  air  resistance  ? 

15.  In  a  tug  of  war  each  team  pulls  with 
a  force  of  2000  Ib.    What  is  the  strain  on 
the  rope  ? 

16.  If  two  men  were  together  in  the  middle  of  a  perfectly  smooth 
(frictionless)  pond  of  ice,  how  could  they  get  off?    Could  one  man  get 
off  if  he  were  there  alone  ? 

17.  If  a  10-g.  bullet  is  shot  from  a  5-kg.  gun  with  a  speed  of  400  m. 
per  second,  what  is  the  backward  speed  of  the  gun  ? 


FIG.  99.  Illustrating  New- 
ton's second  law 


A  laboratory  exercise  on  the  composition  of  forces  should  be  performed  during 
the  study  of  this  chapter.  See,  for  example,  Experiment  11  of  the  authors'  manual. 


CHAPTER  VI 

MOLECULAR  FORCES* 

MOLECULAR  FORCES  IN  SOLIDS.   ELASTICITY 

134.  Proof  of  the  existence  of  molecular  forces  in  solids. 
The  fact  that  a  gas  will  expand  without  limit  as  the  volume 
of  the  containing  vessel  is  increased,  seems  to  show  very  con- 
clusively that  the  molecules  of  gases  do  not  exert  any  appre- 
ciable attractive  forces  upon  one  another.  In  fact,  all  of  the 
experiments  of  Chapter  IV  showed  that  such  substances  cer- 
tainly behave  as  they  would  if  they  consisted  of  independent 
molecules  moving  hither  and  thither  with  great  velocities  and 
influencing  each  other's  motions  only  at  the  instants  of  col- 
lision. Between  collisions  the  molecules  doubtless  move  in 
straight  lines.  It  must  not,  however,  be  thought  that  the  dis- 
tances moved  by  a  single  molecule  between  successive  col- 
lisions are  large.  In  ordinary  air  these  distances  probably  do 
not  average  more  than  .0001  millimeter.  Small,  however,  as 
this  distance  is,  it  is  as  much  as  one  hundred  times  the  radius 
of  a  molecule. 

But  that  the  molecules  of  solids,  on  the  other  hand,  cling 
together  with  forces  of  great  magnitude  is  proved  by  some  of 
the  simplest  facts  of  nature ;  for  solids  not  only  do  not  ex- 
pand indefinitely  like  gases,  but  it  often  requires  enormous 
forces  to  pull  their  molecules  apart.  Thus  a  rod  of  cast  steel 
1  centimeter  in  diameter  may  be  loaded  with  a  weight  of 
7.8  tons  before  it  will  be  pulled  in  two. 

*  This  chapter  should  be  preceded  by  a  laboratory  experiment  in  which  Hooke's 
law  is  discovered  by  the  pupil  for  certain  kinds  of  deformation  easily  measured 
in  the  laboratory.  See,  for  example,  Experiment  13  of  the  authors'  manual. 

101 


102  .K  ...  ..r..:  ,     MOLECULAR  FORCES 

The  following  are  the  weights  in  kilograms  necessary  to 
break  drawn  wires  of  different  materials,  1  square  millimeter 
in  cross  section,  —  the  so-called  relative  tenacities  of  the  wires. 

Lead,  2.6  Copper,  51  Iron,  77 

Silver,  37  Platinum,  43  Steel,  91 

135.  Elasticity.  We  can  obtain  additional  information  about 
the  molecular  forces  existing  in  different  substances  by  study- 
ing what  happens  when  the  weights  applied  are  not  large 

— p —    enough  to  break  the  wires. 

ta 

Thus  let  a  long  steel  wire,  for  example  No.  26  piano  wire, 

be  suspended  from  a  hook  in  the  ceiling,  and  let  the  lower  end 
be  wrapped  tightly  about  one  end  of  a  meter  stick,  as  in  Fig.  100. 
Let  a  fulcrum  c  be  placed  in  a  notch  in  the  stick  at  a  distance 
of  about  5  cm.  from  the  point  of  attachment  to  the  wire,  and  let 
the  other  end  of  the  stick  be  provided  with  a  knitting  needle,  one 
end  of  which  is  opposite  the  vertical  mirror  scale  S.  Let  enough 
weights  be  applied  to  the  pan  P  to  place  the  wire  under  slight 
tension ;  then  let  the  reading  of  the  pointer  p  on  the  scale  S  be 
taken.  Let  3  or  4  kilogram  weights  be  added  successively  to 
the  pan  and  the  corresponding  positions  of  the  pointer  read. 
Then  let. the  readings  be  taken  again  as  the  weights  are  succes- 
sively removed.  In  this  last  operation  the  pointer  will  probably 
be  found  to  come  back  exactly  to  its  first  position. 

This  characteristic  which  the 
steel  has  showTii  in  this  experi- 
ment,  of  returning   to   its   orig- 
inal length  when  the  stretching 
weights  are  removed,  is  an  illus- 
tration  of  a  property  possessed 
Elasticity  of  a  steel     to  a  greater  or  less  extent  by  all 
wire  solid  bodies.  It  is  called  elasticity. 

136.  Limits    of   perfect    elasticity.    If  a  sufficiently  large 
weight  is  applied  to  the  end  of  the  wire  of  Fig.  100,  it  will  be 
found  that  the  pointer  does  not  return  exactly  to  its  original 
position  when  the  weight  is  removed.    We  say,  therefore,  that 


MOLECULAR  FORCES  IN  SOLIDS      103 

steel  is  perfectly  elastic  only  so  long  as  the  distorting  forces 
are  kept  within  certain  limits,  and  that,  as  soon  as  these  limits 
are  overstepped,  it  no  longer  shows  perfect  elasticity.  Differ- 
ent substances  differ  very  greatly  in  the  amount  of  distortion 
which  they  can  sustain  before  they  show  this  failure  to  return 
completely  to  the  original  shape. 

137.  Hooke's  law.  If  we  examine  the  stretches  produced  by 
the  successive  addition  of  kilogram  weights  in  the  experiment 
of  §  135,  Fig.  100,  we  shall  find  that  these  stretches  are  all 
equal,  at  least  within  the  limits  of  observational  error.    Very 
carefully  conducted  experiments  have  shown  that  this  law, 
namely,  that  the  successive  application  of  equal  forces  pro- 
duces a  succession  of  equal  stretches,  holds  very  exactly  for 
all  sorts  of  elastic  displacements,  so  long,  and  only  so  long, 
as  the  limits  of  perfect  elasticity  are  not  overstepped.  1  This 
law  is  known  as  HooMs  law,  after  the  Englishman  Robert 
Hooke  (1635-1703).    Another  way  of  stating  this  law  is  the 
following :    Within  the  limits  of  perfect  elasticity  elastic  deforma-    , 
tions  of  any  sort,  be  they  twists  or  bends  or  stretches,  are  directly  I 
proportional  to  the  forces  producing  them- 

138.  Cohesion   and   adhesion.  The    preceding   experiments 
have  brought  out  the  fact  that  in  the  solid  condition,  at  least, 
molecules  of  the  same  kind  exert  attractive  forces  upon  one 
another.    That   molecules    of   unlike    substances    also    exert 
mutually  attractive  forces  is  equally  true,  as  is  proved  by 
the  fact  that  glue  sticks  to  wood  with  tremendous  tenacity, 
mortar  to  bricks,  nickel  plating  to  iron,  etc. 

The  forces  which  bind  like  kinds  of  molecules  together  are 
commonly  called  cohesive  forces ;  those  which  bind  together 
molecules  of  unlike  kind  are  called  adhesive  forces.  Thus  we 
say  that  mucilage  sticks  to  wood  because  of  adhesion,  while 
wood  itself  holds  together  because  of  cohesion.  Again,  adhe- 
sion holds  the  chalk  to  the  blackboard,  while  cohesion  holds 
together  the  particles  of  the  crayon. 


104  MOLECULAR  FORCES 

139.  Properties  of  solids  depending  on  cohesion.  Many  of  the 
physical  properties  in  which  solid  substances  differ  from  one 
another  depend  on  differences  in  the  cohesive  forces  existing 
between  their  molecules.  Thus  we  are  ac.customed  to  classify 
solids  with  relation  to  their  h  aro1  n  p.siybri  ttlp,n  ftas!^JnctiTity1  m  a,l  - 
j.eability,  tenacity,  elasticity,  .  etc.  The  last  two  of  these  terms 
have  been  sufficiently  explained  in  the  preceding  paragraphs  ; 
but  since  confusion  sometimes  arises  from  failure  to  understand 
the  first  four,  the  tests  for  these  properties  are  here  given. 

We  test  the  relative  hardness  of  two  bodies  by  seeing  which 
will  scratch  the  other.  Thus  the  diamond  is  the  hardest  of  all 
substances,  since  it  scratches  all  others  and  is  scratched  by  none. 

We  test  the  relative  brittleness  of  two  substances  by  seeing 
which  will  break  most  easily  under  a  blow  from  a  hammer.  Thus 
glass  and  ice  are  very  brittle  substances  ;  lead  and  copper  are  not. 

We  test  the  relative  ductility  of  two  bodies  by  seeing  which 
can  be  drawn  into  the  thinner  ^vire.  Platinum  is  the  most  duc- 
tile of  all  substances.  It  has  been  drawn  into  wires  but  .00003 
inch  in  diameter.  Glass  is  also  very  ductile  when  sufficiently 
hot,  as  may  be  readily  shown  by  heating  it  to  softness  in  a 
Bunsen  flame,  when  it  may  be  drawn  into  threads  which  are 
so  fine  as  to  be  almost  invisible. 

We  test  the  relative  malleability  of  two  substances  by  seeing 
which  can  be  hammered  into  the  thinner  sheet.  Gold,  the  most 
malleable  of  all  substances,  has  been  hammered  into  sheets 
inch  in  thickness. 


QUESTIONS  AND  PROBLEMS 

1.  Why  atre  springs  made  of  steel  ratker  than  of  copper? 

2.  If  a  given  weight  is  required  to  break  a  given  wire,  how  much 
force  is  required  to  break  two  such  wires  hanging  side  by  side  ?    How 
much  to  break  one  wire  of  twice  the  diameter? 

3.  What  must  be  the  cross  section  of  a  wire  of  copper  if  it  is  to  have 
the  same  tensile  strength  (that  is,  break  with  the  same  weight)  as  a 
wire  of  iron  1  sq.  mm.  in  cross  section  ? 


MOLECULAK  FOECES  IK  LIQUIDS 


105 


4.  How  many  times  greater  must  the  diameter  of  one  wire  be  than 
that  of  another  of  the  same  material  if  it  is  to  have  five  times  the  tensile 
strength  ? 

5.  If  the  position  of  the  pointer  on  a  spring  balance  is  marked  when 
no  load  is  on  the  spring,  and  again  when  the  spring  is  stretched  with  a 
load  of  10  g.,  and  if  the  space  between  the  two  marks  is  then  divided 
into  ten  equal  parts,  will  each  of  these  parts  represent  a  gram  ?   Why  ? 


FIG.  101.    Illustrating 
cohesion  of  water 


MOLECULAR  FORCES  IN  LIQUIDS.   CAPILLARY  PHENOMENA 

140.  Proof  of  the  existence  of  molecular  forces  in  liquids. 

The  facility  with  which  liquids  change  their  shape  might  lead 
us  to  suspect  that  the  molecules  of  such  substances  exert 
almost  no  forces  upon  one  another,  but 
a  simple  experiment  will  show  that  this 
is  far  from  true. 

By  means  of  sealing  wax  and  string  let  a 
glass  plate  be  suspended  horizontally  from  one 
arm  of  a  balance,  as  in  Fig.  101.  After  equilib- 
rium is  obtained  let  a  surface  of  water  be  placed 
just  beneath  the  plate  and  the  beam  pushed 
down  until  contact  is  made.  It  will  be  found 
necessary  to  add  a  considerable  weight  to  the 
opposite  pan  in  order  to  pull  the  plate  away  from  the  water.  Since  a 
layer  of  water  will  be  found  to  cling  to  the  glass,  it  is  evident  that  the 
added  force  applied  to  the  pan  has  been  expended  in  pulling  water' 
molecules  away  from  water  molecules,  not  in  pulling  glass  away  from 
water.  Similar  experiments  may  be  performed  with  all  liquids.  In  the 
case  of  mercury  the  glass  will  not  be  found  to  be  wet,  showing  that  the 
cohesion  of  mercury  is  greater  than  the  adhesion  of  glass  and  mercury. 

141.  Shape  assumed  by  a  free  liquid.    Since,  then,  every 
molecule  of  a  liquid  is  pulling  on  every  other  molecule,  any 
body  of  liquid  which  is  free  to  take  its  natural  shape,  that  is, 
which  is  acted  on 'only  by  its  own  cohesive  forces,  must  draw 
itself  together  until  it  has  the  smallest  possible  surface  com- 
patible with  its  volume  ;  for,  since  every  molecule  in  the  surface 
is  drawn  toward  the  interior  by  the  attraction  of  the  molecules 


106  MOLECULAK  FOBCES 

within,  it  is  clear  that  molecules  must  continually  move  toward 
the  center  of  the  mass  until  the  whole  has  reached  the  most 
compact  form  possible.  Now  the  geometrical  figure  which  has 
the  smallest  area  for  a  given  volume  is  a  sphere.  We  conclude, 
therefore,  that  if  we  could  relieve  a  body  of  liquid  from  the 
action  of  gravity  and  other  outside  forces,  it  would  at  once 
take  the  form  of  a  perfect  sphere.  This  conclusion  may  be 
easily  verified  by  the  following  experiment : 

Let  alcohol  be  added  to  water  until  a  solution  is  obtained  in  which 
a  drop  of  common  lubricating  oil  will  float  at  any  depth.  Then  with  a 
pipette  insert  a  large  globule  of  oil  beneath  the  surface.  The  oil  will  be 

seen  to  float  as  a  perfect  sphere  within  the  body  

of  the  liquid  (Fig.  102).  (Unless  the  drop  is 
viewed  from  above,  the  vessel  should  have 
flat  rather  than  cylindrical  sides,  otherwise  the 
curved  surface  of  the  water  will  act  like  a  lens 

and  make  the  drop  appear  flattened.)  FIG.  102.     Spherical 

J    _ .       .  ,  globule  of  oil,  freed 

The  reason  that  liquids  are  not  more     from  action  of  gravity 

commonly  observed  to  take  the  spherical 
form  is  that  ordinarily  the  force  of  gravity  is  so  large  as  to 
be  more  influential  in  determining  their  shape  than  are  the 
cohesive  forces.  As  verification  of  this  statement  we  have 
only  to  observe  that  as  a  body  of  liquid  becomes  smaller 
and  smaller,  —  that  is,  as  the  gravitational  forces  upon  it 
become  less  and  less,  —  it  does  indeed  tend  more  and  more 
to  take  the  spherical  form.  Thus  very  small  globules  of  mer- 
cury on  a  table  will  be  found  to  be  almost  perfect  spheres, 
and  raindrops  or  minute  floating  particles  of  all  liquids  are 
quite  accurately  spherical. 

142.  Contractility  of  liquid  films ;  surface  tension.  The  tend- 
ency of  liquids  to  assume  the  smallest  possible  surface  fur- 
nishes a  simple  explanation  of  the  contractility  of  liquid  films. 

Let  a  soap  bubble  2  or  3  inches  in  diameter  be  blown  on  the  bowl 
of  a  pipe  and  then  allowed  to  stand.  It  will  at  once  begin  to  shrink 
in  size  and  in  a  few  minutes  will  disappear  within  the  bowl  of  the  pipe. 


MOLECULAR  FOBCES  IN  LIQUIDS  10T 

The  liquid  of  the  bubble  is  simply  obeying  the  tendency  to  reduce  its 
surface  to  a  minimum,  a  tendency  which  is  due  only  to  the  mutual  at- 
tractions which  its  molecules  exert  upon  one  another.  A  candle  flame 
held  opposite  the  opening  in  the  stem  of  the  pipe  will  be  deflected  by 
the  current  of  air  which  the  contracting  bubble  is  forcing  out  through 
the  stem. 

Again,  let  a  loop  of  fine  thread  be  tied  to  the  edge  of  a  wire  ring,  as 
in  Fig.  103.  Let  the  ring  be  dipped  into  a  soap  solution  so  as  to  form  a 
film  across  it,  and  then  let  a  hot  wire  be  thrust  through  the  film  inside 
the  loop.  The  tendency  of  the  film  outside  of  the  loop  to  contract  will  in- 
stantly snap  out  the  thread  into  a  perfect  circle  (Fig.  104).  The  reason 
that  the  thread  takes  the  circular  form  is  that  since  the  film  outside  the 


FIG.  103  FIG.  104  FIG.  105 

Illustrating  the  contractility  of  soap  films 

loop  is  striving  to  assume  the  smallest  possible  surface,  the  area  inside 
the  loop  must  of  course  become  as  large  as  possible.  The  circJe  is  the 
figure  which  has  the  largest  possible  area  for  a  ^.ven  perimeter. 

Let  a  soap  film  be  formed  across  the  mouth  of  a  clean  2-inch  funnel, 
as  in  Fig.  105.  The  tendency  of  the  film  to  contract  will  be  sufficient 
to  lift  its  weight  against  the  force  of  gravity. 

The  tendency  of  a  liquid  to  reduce  its  exposed  surface  to  a 
minimum,  that  is,  the  tendency  of  any  liquid  surface  to  act  like 
a  stretched  elastic  membrane  is  called  surface  tension. 

143.  Ascension  and  depression  of  liquids  in  capillary  tubes. 
It  was  shown  in  Chapter  II  that,  in  general,  a  liquid  stands 
at  the  same  level  in  any  number  of  communicating  vessels. 
The  following  experiments  will  show  that  this  rule  ceases  to 
hold  in  the  case  of  tubes  of  small  diameter. 


108 


MOLECULAR  FORCES 


FIG.  106.     Kise   of 

liquids  in  capillary 

tubes 


Let  a  series  of  capillary  tubes  of  diameter  varying  from  2  mm.  to  .1  mm. 
be  arranged  as  in  Fig.  106. 

When  water  is  poured  into  the  vessel  it  will  be  found  to  rise  higher 
in  the  tubes  than  in  the  vessel,  and  it  will  be  seen  that  the  smaller 
the  tube  the  greater  the  height  to  which  it  rises. 
If  the  water  is  replaced  by  mercury,  howev.er,  the 
effects  will  be  found  to  be  just  inverted.  The  mer- 
cury is  depressed  in  all  the  tubes,  the  depression 
being  greater  in  proportion  as  the  tube  is  smaller 
[Fig.  107,  (1)].  This  depression  is  most  easily  ob- 
served with  a  U-tube  like  that  shown  in  Fig.  107,  (2). 

Experiments  of  this  sort  have  established 
the  following  laws : 

1.  Liquids  rise  in  capillary  tubes  when  they 
are  capable  of  wetting  them,  but  are  depressed 
in  tubes  which  they  do  not  wet. 

2.  The  elevation  in  the  one  case  and  the  depression  in  the 
other  are  inversely  proportional  to  the  diameters  of  the  tubes. 

It  will  be  noticed,  too,  that  when  a  liquid  rises,  its  surface 
within  the  tube  is  concave  upward,  and  when  it  is  depressed 
its  surface  is  convex  upward. 

144.  Cause  of  curvature  of 
a  liquid  surface  in  a  capillary 
tube.  All  of  the  effects  pre- 
sented in  the  last  paragraph 
can  be  explained  by  a  con- 
sideration of  cohesive  and 
adhesive  forces.  However, 
throughout  the  explanation 
we  must  keep  in  mind  two 
familiar  facts :  first,  that  the  surface  of  a  body  of  water  at  rest, 
for  example  a  pond,  is  at  right  angles  to  the  resultant  force, 
that  is,  gravity,  which  acts  upon  it ;  and  second,  that  the  force 
of  gravity  acting  on  a  minute  amount  of  liquid  is  negligible  in 
comparison  with  its  own  cohesive  force  (see  §  141). 


FIG.  107.    Depression  of  mercury  in 
capillary  tubes 


MOLECULAE  FOECES  IN  LIQUIDS 


109 


Consider,  then,  a  very  small  body  of  liquid  close  to  the 
point  o  (Fig.-  108),  where  water  is  in  contact  with  the  glass 
wall  of  the  tube.  Let  the  quantity  of  liquid  considered  be 
so  minute  that  the  force  of  gravity  acting  upon  it  may  be 
disregarded.  The  force,  of  adhesion  of  the  wall  will  pull  the 


FIG.  108  FIG.  109 

Condition  for  elevation  of  a  liquid  near  a  wall 

liquid  particles  at  o  in  the  direction  oE.  The  force  of  cohesion 
of  the  liquid  will  pull  these  same  particles  in  the  direction  oF. 
The  resultant  of  these  two  pulls  on  the  liquid  at  o  will  then 
be  represented  by  oR  (Fig.  108),  in  accordance  with  the  paral- 
lelogram law  of  Chapter  V.  If,  then,  the  adhesive  force  oE 
exceeds  the  cohesive  force  of\  the  direc- 
tion oR  of  the  resultant  force  will  lie  to 
the  left  of  the  vertical  om  (Fig.  109), 
in  which  case,  since  the  surface  of  a 
liquid  always  assumes  a  position  at 
right  angles  to  the  resultant  force,  it 
must  rise  up  against  the  wall  as  water 
does  against  glass  (Fig.  109). 

If  the  cohesive  force  oF  (Fig.  110)  is 
strong  in  comparison  with  the  adhesive 

force  oE,  the  resultant  oR  will  fall  to  the  right  of  the  ver- 
tical, in  which  case  the  liquid  must  be  depressed  about  o. 

Whether,  then,  a  liquid  will  rise  against  a  solid  wall  or  be 
depressed  by  it  will  depend  only  on  the  relative  strengths  of 
the  adhesion  of  the  wall  for  the  liquid  and  the  cohesion  of  the 


FIG.  110.    Condition   for 

the  depression  of  a  liquid 

near  a  wall 


110 


MOLECULAR  FORCES 


liquid  for  itself.  Since  mercury  does  not  wet  glass,  we  know 
that  cohesion  is  here  relatively  strong,  and  we  should  expect, 
therefore,  that  the  mercury  would  be  depressed,  as  indeed  we 
find  it  to  be.  The  fact  that  water  will  wet  glass  indicates  that 
in  this  case  adhesion  is  relatively  strong,  and  hence  we  should 
expect  water  to  rise  against  the  walls  of  the  containing  vessel, 
as  in  fact  it  does. 

It  is  clear  that  a  liquid  which  is  depressed  near  the  edge 
of  a  vertical  solid  wall  must  assume  within  a  tube  a  surface 
which  is  convex  upward,  while  a  liquid  which  rises  against  a 
wall  must  within  such  a  tube  be  concave  upward. 

145.  Explanation  of  ascension  and  depression  in  capillary 
tubes.  As  soon  as  the  curvatures  just  mentioned  are  produced, 
the  concave  surface  aob  (Fig.  Ill)  tends,  by  virtue  of  surface 


FIG.  Ill  FIG.  112 

A  concave  meniscus  causes  a  rise 
in  capillary  tube 


•-• 


FIG.  113 


FIG.  114 


A  convex  meniscus  causes 
a  fall 


tension,  to  straighten  out  into  the  flat  surface  ao'b.  But  it  no 
sooner  thus  begins  to  straighten  out  than  adhesion  again  ele- 
vates it  at  the  edges.  It  will  be  seen,  therefore,  that  the  liquid 
must  continue  to  rise  in  the  tube  until  the  weight  of  the  vol- 
ume of  liquid  lifted,  namely  amnb  (Fig.  112),  balances  the 
tendency  of  the  surface  aob  to  flatten  out.  That  the  liquid 
will  rise  higher  in  a  small  tube  than  in  a  large  one  is  to  be 
expected,  since  the  weight  of  the  column  of  liquid  to  be  sup- 
ported in  the  small  tube  is  less. 


MOLECULAE  FOECES  IN  LIQUIDS  111 

Similarly,  the  convex  mercury  surface  aob  (Fig.  113)  falls 
until  the  upward  pressure  at  0,  due  to  the  depth  h  of  mer- 
cury (Fig.  114),  balances  the  tendency  of  the  surface  aob  to 
flatten  out. 

146.  Capillary  phenomena  in  everyday  life.    Capillary  phe- 
nomena play  a  very  important  part  in  the  processes  of  nature 
and  of  everyday  life.    Thus  the  rise  of  oil  in  wicks  of  lamps, 
the  complete  wetting  of  a  towel  when  one  end  of  it  is  allowed 
to  stand  in  a  basin  of  water,  the  rapid  absorption  of  liquid  by 
a  lump  of  sugar  when  one  corner  of  it  only  is  immersed,  the 
taking  up  of  ink  by  blotting  paper,  are  all  illustrations  of  pre- 
cisely the  same  phenomena  which  we  observe  in  the  capillary 
tubes  of  Fig.  106. 

147.  Floating  of  small  objects  on  water.    Let  a  needle  be  laid 
very  carefully  on  the  surface  of  a  dish  of  water.    In  spite  of  the  fact 
that  it  is  nearly  eight  times  as  dense  as  water  it  will 

be  found  to  float.  If  the  needle  has  been  previously 
magnetized,  it  may  be  made  to  move  about  in  any 
direction  over  the  surface  in  obedience  to  the  pull  of 
a  magnet  held,  for  example,  underneath  the  table.  FIG.  115.  Cross 

,      .  section     of    a 

lo  discover  the  cause  of  this  apparently  im-    floating  needle 

possible  phenomenon,  examine  closely  the  sur- 
face of  the  water  in  the  immediate  neighborhood  of  the 
needle.  It  will  be  found  to  be  depressed  in  the  manner 
shown  in  Fig.  115.  This  furnishes  at  once  the  explanation. 
So  long  as  the  needle  is  so  small  that  its  own  weight  is  no 
greater  than  the  upward  force  exerted  upon  it  by  the  tend- 
ency of  the  depressed  (and  therefore  concave)  liquid  surface 
to  straighten  out  into  a  flat  surface,  the  needle  could  not 
sink  in  the  liquid,  no  matter  how  great  its  density.  If  the 
water  had  wet  the  needle,  -that  is,  if  it  had  risen  about 
the  needle  instead  of  being  depressed,  the  tendency  of  the 
liquid  surface  to  flatten  out  would  have  pulled  it  down  into 
the  liquid  instead  of  forcing  it  upward.  Any  body  about 


112  MOLECULAR  FORCES 

which  a  liquid  is  depressed  will  therefore  float  on  the  surface 
of  the  liquid  if  its  mass  is  not  too  great.  Even  if  the  liquid 
tends  to  rise  about  a  body  when  it  is  perfectly  clean,  an  im- 
perceptible film  of  oil  upon  the  body 
Avill  cause  it  to  depress  the  liquid,  and 
hence  to  float. 

The  above  experiment  explains  the  TIG.  116.  Insect  walking 
familiar  phenomenon  of  insects  walk- 
ing and  running  on  the  surface  of  water  (Fig.  116)  in  appar- 
ent contradiction  to  the  law  of  Archimedes,  in  accordance 
with  which  they  should  sink  until  they  displace  their  own 
weight  of  the  liquid. 

• 

QUESTIONS  AND  PROBLEMS 

1.  Shot  are  made  by  pouring  molten  lead  through  a  sieve  on  top 
of  a  tall  tower  and  catching  it  in  water  at  the  bottom.  Why  are  they 
spherical  ? 

2.  Would  mercury  ascend  a  lamp  wick  as  oil  and  water  do? 

3.  If  water  will  rise  32  cm.  in  a  tube  .1  mm.  in  diameter,  how  high 
will  it  rise  in  a  tube  .01  mm.  in  diameter? 

4.  Candle  grease  may  be  removed  from  clothing  by  covering  it 
with  blotting  paper  and  then  passing  a  hot  flatiron  over  the  paper. 
Explain. 

5.  Why  does  a  small  stream  of  water  break  up  into  drops  instead 
of  falling  as  a  continuous  thread? 

6.  Why  will  a  piece  of  sharp-cornered  glass  become  rounded  when 
heated  to  redness  in  a  Bunsen  flame  ? 

7.  The  leads  for  pencils  are  made  by  subjecting  powdered  graphite 
to  enormous  pressures  produced  by  hydraulic  machines.    Explain  how 
the  pressure  changes  the  powder  to  a  coherent  mass. 

8.  Float  two  matches  an  inch  apart.    Touch  the  water  between 
them  with  a  hot  wire.    The  matches  will  spring  apart.   What  does  this 
show  about  the  effect  of  temperature  on  surface  tension  ? 

9.  Repeat  the  experiment,  touching  the  water  with  a  wire  moistened 
with  alcohol.    What  do  you  infer  as  to  the  relative  surface  tensions  of 
alcohol  and  water? 

10.   Rub  a  little  soap  on  one  end  of  half  a  toothpick  and  lay  it  upon  the 
surface  of  a  large  vessel  of  clean  still  water.  Explain  the  observed  motion. 


ABSORPTION  OF  GASES 


113 


ABSORPTION  OF  GASES  BY  SOLIDS  AND  LIQUIDS 

148.  Absorption  Of  gases  by  SOlids.  Let  a  large  test  tube  be 
filled  with  ammonia  gas  by  heating  aqua  ammonia  and  causing  the 
evolved  gas  to  displace  mercury  in  the  tube,  as  in  Fig.  117.  Let  a 
piece  of  charcoal  an  inch  long 
and  nearly  as  wide  as  the  tube 
be  heated  to  redness  and  then 
plunged  beneath  the  mercury. 
When  it  is  cool  let  it  be  slipped 
underneath  the  mouth  of  the 
test  tube  and  allowed  to  rise  into 
the  gas.  The  mercury  will  be 
seen  to  rise  in  the  tube,  as  in 


FIG.  117.    Filling  tube  with  ammonia 


Fig.  118,  thus  showing  that  the  gas  is  being  absorbed  by  the  charcoal. 
If  the  gas  is  unmixed  with  air,  the  mercury  will  rise  to  the  very  top  of 
the  tube,  thus  showing  that  all  the  ammonia  has  been  absorbed  by 
the  charcoal. 

This  property  of  absorbing  gases  is  possessed  to  a  notable 
degree  by  porous  substances,  such  as  meerschaum,  gypsum, 
charcoal,  etc.,  especially  coconut  charcoal.  It 
is  not  improbable  that  all  solids  hold,  closely 
adhering  to  their  surfaces,  thin  layers  of  the 
.gases  with  which  they  are  in  contact,  and  that 
the  prominence  of  the  phenomena  of  absorp- 
tion in  porous  substances  is  due  to  the  great 
extent  of  surface  possessed  by  such  substances. 

That   the    same    substance   exerts  widely 

different  attractions  upon  the  molecules  of 

TXV  i  i         ,1        (-          ,i  FIG.  118.  Absorp- 

different   gases   is   shown    by  the  fact  that     tion  of  ammonia 

charcoal  will  absorb  90  times  its  own  volume      gas  by  charcoal 
of  ammonia  gas,  35  times  its  volume  of  car- 
bon dioxide,  and  but  1.7  times  its  volume  of  hydrogen.    The 
usefulness  of  charcoal  as  a  deodorizer  is  due  to  its  enormous 
ability  to  absorb  certain  kinds  of  gases. 

149,   Absorption  of   gases  in  liquids.     Let  a  beaker  containing 
cold  water  be  slowly  heated.    Small  bubbles  of  air  will  be  seen  to  collect 

t 


114 


MOLECULAR  FORCES 


FIG.  119.   Absorp- 
tion   of    ammonia 
by  water 


in  great  numbers  upon  the  walls  and  to  rise  through  the  liquid  to  the 
surface.  That  they  are  indeed  bubbles  of  air  and  not  of  steam  is  proved 
first  by  the  fact  that  they  appear  when  the  temper- 
ature is  far  below  boiling,  and  second  by  the  fact  that 
they  do  not  condense  as  they  rise  into  the  higher 
and  cooler  layers  of  the  water. 

The  experiment  shows  two  things:  first, 
that  water  ordinarily  contains  considerable 
quantities  of  air  dissolved  in  it ;  and  second, 
that  the  amount  of  air  which  water  can  hold 
decreases  as  the  temperature  rises.  The  first 
point  is  also  proved  by  the  existence  of  fish 
life;  for  fishes  obtain  the  oxygen  which  they 
need  to  support  life,  not  immediately  from  the 
water,  but  from  the  air  which  is  dissolved  in  it. 

The  amount  of  gas  which  will  be  absorbed 
by  water  varies  greatly  with  the  nature  of  the 
gas.  At  0°  C.  and  a  pressure  of  76  centimeters 
1  cubic  centimeter  of  water  will  absorb  1050  cubic  centi- 
meters of  ammonia,  1.8  cubic  centimeters  of  carbon  dioxide, 
and  but  .04  cubic  centimeter  of  oxygen.  Ammonia  itself  is  a 
gas  under  ordinary  conditions.  The  commercial  aqua  ammonia 
is  simply  ammonia  gas  dissolved  in  water. 

The  following  experiment  illustrates  the  absorption  of 
ammonia  by  water : 

Let  the  flask  F  (Fig.  110)  and  tube  b  be  filled  with  ammonia  by  passing 
a  current  of  the  gas  in  -at  a  and  out  through  It.  Then  let  a  be  corked 
up  and  b  thrust  into  G,  a  flask  nearly  filled  with  water  which  has  been 
colored  slightly  red  by  the  addition  of  litmus  and  a  drop  or  two  of  acid. 
As  the  ammonia  is  absorbed  the  water  will  slowly  rise  in  &,  and  as  soon 
as  it  reaches  F  it  will  rush  up  very  rapidly  until  the  upper  flask  is 
nearly  full.  At  the  same  time  the  color  will  change  from  red  to  blue 
because  of  the  action  of  the  ammonia  upon  the  litmus. 

Experiment  shows  that  in  every  case  of  absorption  of  a  gas 
by  a  liquid  or  a  solid,  the  quantity  of  gas  absorbed  decreases  ivith 


ABSORPTION  OF  GASES  115 

an  increase  in  temperature  —  a  result  which  was  to  have  been 
expected  from  the  kinetic  theory,  since  increasing  the  molec- 
ular velocity  must  of  course  increase  the  difficulty  which  the 
adhesive  forces  have  in  retaining  the  gaseous  molecules. 

150.  Effect  of  pressure  upon  absorption.  Soda  water  is  ordi- 
nary water  which  has  been  made  to  absorb  large  quantities  of 
carbon  dioxide  gas.  This  impregnation  is  accomplished  by 
bringing  the  water  into  contact  with  the  gas  under  high 
pressure.  As  soon  as  the  pressure  is  relieved  the  gas  passes 
rapidly  out  of  solution.  This  is  the  cause  of  the  characteristic 
effervescence  of  soda  water.  These  facts  show  clearly  that  the 
amount  of  carbon  dioxide  which  can  be  absorbed  by  water  is 
greater  for  high  pressures  than  for  low.  As  a  matter  of  fact, 
careful  experiments  have  shown  that  the  amount  of  any  gas 
absorbed  is  directly  proportional  to  the  pressure,  so  that  if 
carbon  dioxide  under  a  pressure  of  10  atmospheres  is  brought 
into  contact  with  water,  ten  times  as  much  of  the  gas  is  ab- 
sorbed as  if  it  had  been  under  a  pressure  of  1  atmosphere. 

QUESTIONS  AND  PROBLEMS 

1.  Capillary  action  is  much  more  effective  in  bringing  moisture  to  the 
surface  in  tightly  packed  soil  than  in  loose  soil  where  the  spaces  between 
the  earth  particles  are  much  greater.  Why,  then,  is  it  advantageous  to 
crops  to  keep  the  surface  loose  (dry  fanning)  ? 

2.  Why  do  fishes  in  an  aquarium  die  if  the  water  is  not  frequently 
renewed  ? 

3.  Explain   the  apparent  generation  of   ammonia  gas  when  aqua 
ammonia  is  heated. 

4.  Why  in  the  experiment  illustrated  in  Fig.  119  was  the  flow  so  much 
more  rapid  after  the  water  began  to  run  over  into  F  ? 

5.  How  can  you  tell  whether  bubbles  which  rise  from  the  bottom  of 
a  vessel  which  is  being  heated  are  bubbles  of  air  or  bubbles  of  steam  ? 


CHAPTER  VII 

THERMOMETRY ;  EXPANSION  COEFFICIENTS* 
THERMOMETRY 

151.  Meaning  of  temperature.   When  a  body  feels  hot  to  the 
touch  we  are  accustomed  to  say  that  it  has  a  high  temperature, 
and  when  it  feels  cold  that  it  has  a  low  temperature.   Thus  the 
word  "  temperature  "  is  used  to  denote  the  condition  of  hotness 
or  coldness  of  the  body  whose  state  is  being  described. 

152.  Measurement  of  temperature.   So  far  as  we  know,  up  to 
the  time  of  Galileo  no  one  had  ever  used  any  special  instrument 
for  the  measurement  of  temperature.   People  knew  how  hot  or 
how  cold  it  was  from  their  feelings  only.    But  under  some  con- 
ditions this  temperature  sense  is  a  very  unreliable  guide.   For 
example,  if  the  hand  has  been  in  hot  water,  tepid  water  will 
feel  cold ;  while  if  it  has  been  in  cold  water,  the  same  tepid 
water  will  feel  warm  ;  a  room  may  feel  hot  to  one  who  has  been 
running,  while  it  will  feel  cool  to  one  who  has  been  sitting  still. 

Difficulties  of  this  sort  have  led  to  the  introduction  in 
modern  times  of  mechanical  devices,  called  thermometers,  for 
measuring  temperature.  These  instruments  depend  for  their 
operation  upon  the  fact  that  practically  all  bodies  expand  as 
they  grow  hot. 

153.  Galileo's  thermometer.    It  was  in  1592  that  Galileo, 
at  the  University  of  Padua  in  Italy,   constructed  the  first 
thermometer.    He  was  familiar  with  the  facts  of  expansion 

*  It  is  recommended  that  this  chapter  be  preceded  by  laboratory  measure- 
ments on  the  expansions  of  a  gas  and  a  solid.  See,  for  example,  Experiments  14 
and  15  of  the  authors'  manual. 

116 


THERMOMETKY 


117 


FIG.  120.    Galileo's 
thermometer 


of  solids,  liquids,  and  gases ;  and  since  gases  expand  more 
than  solids  or  liquids,  he  chose  a  gas  as  his  expanding 
substance.  His  device  was  that  shown  in 
Fig.  120. 

The  relative  hotness  of  two  bodies  was 
compared  by  observing  which  one  of  the 
two,  when  placed  in  contact  with  the  air 
bulb,  caused  the  liquid  to  descend  farther 
in  the  stem  S.  As  a  matter  of  fact,  baro- 
metric as  well  as  temperature  changes  cause 
changes  in  the  height  of  the  liquid  in  the 
stern  of  such  an  instrument,  but  Galileo  does 
not  seem  to  have  been  aware  of  this  fact. 

It  was  not  until  about^  1 700  that  mercury  thermometers 
were  invented.  On  account  of  their  extreme  convenience 
these  have  now  replaced  all  others  for  prac- 
tical purposes. 

154.  The  construction  of  a  centigrade  mer- 
cury thermometer.  The  meaning  of  a  degree 
of  temperature  change  is  best  understood 
from  a  description  of -the  method  of  making 
and  graduating  a  mercury  thermometer. 

A  bulb  is  blown  at  one  end  of  a  piece 
of  thick-walled  glass  tubing  of  small,  uni- 
form bore.  Bulb  and  tube  are  then  filled  with 
mercury,  at  a  temperature  slightly  above  the 
highest  temperature  for  which  the  thermom- 
eter is  to  be  used,  and  the  tube  is  sealed 
off  in  a  hot  flame.  As  the  mercury  cools,  it  of  finding  the  0° 

contracts  and  falls  away  from  the  top  of  the    point  of  a  ther~ 

J  mometer 

tube,  leaving  a  vacuum  above  it. 

The  bulb  is  next  surrounded  with  melting  snow  or  ice,  as  in 
Fig.  121,  and  the  point  at  which  the  mercury  stands  in  the  tube 
is  marked  0°.  Then  the  bulb  and  tube  are  placed  in  the  steam 


FIG.  121.  Method 


118     THERMOMETRY ;  EXPANSION  COEFFICIENTS 


100" 


rising  from  boiling  water,  as  in  Fig.  122,  and  the  new  position 
of  the  mercury  is  marked  100°.  The  space  between  these  two 
marks  on  the  stem  is  then  divided  into  100  equal  parts,  and 
divisions  of  the  same  length  are  extended 
above  the  100°  mark  and  below  the  0°  mark. 

One  degree  of  change  in  temperature,  meas- 
ured on  such  a  thermometer,  means,  then, 
such  a  temperature  change  as  will  cause  the 
mercury  in  the  stem  to  move  over  one  of 
these  divisions ;  that  is,  it  is  such  a  tempera- 
ture change  as  will  cause  mercury  contained 
in  a  glass  bulb  to  expand  y^-  of  the  amount 
which  it  expands  in  passing  from  the  temper- 
ature of  melting  ice  to  that  of  boiling  water. 
A  thermometer  in  which  the  scale  is  divided  in 
this  way  is  called  a  centigrade  thermometer. 

Thermometers  graduated  on  the  centigrade 
scale  are  used  almost  exclusively  in  scien- 
tific work,  and  also  for  ordinary  purposes  in 
most  countries  which  have  adopted  the  metric 
system.  This  scale  was  first  devised  in  1742 
by  Celsius,  of  Upsala,  Sweden.  For  this  reason  it  is  some- 
times called  the  Celsius  instead  of  the  centigrade  scale. 

155.  Fahrenheit  thermometers.  The  common  household 
thermometer  in  England  and  the  United  States  differs  from 
the  centigrade  only  in  the  manner  of  its  graduation.  In  its 
construction  the  temperature  of  melting  ice  is  marked  32° 
instead  of  0°,  and  that  of  boiling  water  212°  instead  of  100°. 
The  intervening  stem  is  then  divided  into  180  parts.  The 
zero  of  this  scale  is  the  temperature  obtained  by  mixing  equal 
weights  of  sal  ammoniac  (ammonium  chloride)  and  snow.  In 
1714,  when  Fahrenheit,  of  Danzig,  Germany,  devised  this  scale, 
he  chose  this  zero  because  he  thought  it  represented  the  lowest 
possible  temperature,  that  is,  the  entire  absence  of  heat. 


FIG.  122.    Method 
of  finding  the  100° 
point    of   a   ther- 
mometer 


THERMOMETRY 


119 


100° 

90* 


60° 
50° 
40° 
30° 
20° 
10° 

o" 

-10* 


156.  Comparison  of  centigrade  and  Fahrenheit  thermome- 
ters.   From  the  methods  of  graduation   of  the  Fahrenheit 
and  centigrade  thermometers  it  will  be  seen  that  1.00°  on  the 
centigrade  scale  denotes  the  same  difference  c     F 

of  temperature  as  180°  on  the  Fahrenheit 
scale  (Fig.  123).  Hence  ohe  Fahrenheit  de- 
gree is  equal  to  five  ninths  of  a  centigrade 
degree,  and  one  centigrade  degree  is  equal 
to  nine  fifths  of  a  Fahrenheit  degree.  Hence 
to  reduce  from  the  Fahrenheit  to  the  centi- 
grade scale,  first  find  how  many  Fahrenheit 
degrees  the  given  temperature  is  above  or  below 
the  freezing  temperature,  and  then  multiply  by 
five  ninths. 

To  reduce  from  centigrade  to  Fahrenheit, 
first  multiply  by  nine  fifths  in  order  to  find  how 
many  Fahrenheit  degrees  the  given  tempera- 
ture is  above  or  below  the  freezing  temperature. 
Knowing  how  far  it  is  from  the  freezing  point, 
it  will  be  very  easy  to  find  how  far  it  is  from 
0°  F.,  which  is  32°  below  the  freezing  point. 

157.  The  range  of  the  mercury  thermometer.    Since  mer- 
cury freezes  at  —  39°  C.,  temperatures  lower  than  this  are  very 
often  measured  by  means  of  alcohol  thermometers,  for  the  freez- 
ing point  of  alcohol  is  — 130°  C.    Similarly,  since  the  boiling 
point  of  mercury  is  360°  C.,  mercury  thermometers  cannot  be 
used  for  measuring  very  high  temperatures.    For  both  very 
high  and  very  low  temperatures,  in  fact  for  all  temperatures, 
a  gas  thermometer  is  the  standard  instrument. 

158.  The  standard  hydrogen  thermometer.    The  modern  gas 
thermometer  (Fig.  124)  is,  however,  widely  different  from 
that  devised  by  Galileo  (Fig.  120).    It  is  not  usually  the  in- 
crease in  the  volume  of  a  gas  kept  under  constant  pressure 
which  is  taken  as  the  measure  of  temperature  change,  but 


1 194" 
176° 
158" 
140° 
122° 
\104° 
86° 
68" 
50° 
32° 
14° 
0° 


FIG.  123.    The  cen- 
tigrade and  Fahren- 
heit scales 


120     THERMOMETRY;  EXPANSION  COEFFICIENTS 


rather  the  increase  in  pressure  which  the  molecules  of  a 
confined  gas  exert  against  the  walls  of  a  vessel  whose  vol- 
ume is  kept  constant.  The  essential  features  of  the  method  of 
calibration  and  use  of  the  standard  hydrogen  thermometer  at 
the  International  Bureau  of  Weights  and 
Measures  at  Paris  are  as  follows : 

The  bulb  B  (Fig.  124)  is  first  filled  with  hydro-  ioo°C 

gen  and  the  space  above  the  mercury  in  the  tube  a 
made  as  nearly  a  perfect  vacuum  as  possible.  B  is 
then  surrounded  with  melting  ice  (as  in  Fig.  121) 
and  the  tube  a  raised  or  lowered  until  the  mer- 
cury in  the  arm  b  stands  exactly  opposite  the  fixed 
mark  c  on  the  tube.  Now,  since  the  space  above  D 
is  a  vacuum,  the  pressure  exerted  by  the  hydrogen 
in  B  against  the  mercury  surface  at  c  just  sup- 
ports the  mercury  column  ED.  The  point  D  is 
marked  on  a  strip  of  metaTHoehind  the  tube  a. 
The  bulb  B  is  then  placed  in  a  steam  bath  like 
that  shown  in  Fig:  122.  The  increased  pressure 
of  the  gas  in  B  at  once  begins  to  force  the  mer- 
cury down  at  c  and  up  at  D.  But  by  raising  the 
slrm  a  the  mercury  in  b  is  forced  back  again  to  c, 
the  increased  pressure  of  the  gas  on  the  surface 
of  the  mercury  at  c  being  balanced  by  the  in- 
creased height  of  the  mercury  column  supported, 
which  is  now  EF  instead  of  ED.  When  the  gas 
in  B  is  thoroughly  heated  to  the  temperature  of 
the  steam,  the  arm  a  is  very  carefully  adjusted 
so  that  the  mercury  in  b  stands  very  exactly  at  c, 
its  original  level.  A  second  mark  is  then  placed  on 
the  metal  strip  exactly  opposite  the  new  level  of  the  mercury,  that  is, 
at  F.  D  is  then  marked  0°C.,  and  F  is  marked  100°  C.  The  vertical 
distance  between  these  marks  is  divided  into  100  exactly  equal  parts. 
Divisions  of  exactly  the  same  length  are  carried  above  the  100°  mark 
and  below  the  0°  mark.  One  degree  of  change  in  temperature  is  then 
defined  as  any  change  in  temperature  which  will  cause  the  pressure 
of  the  gas  in  B  to  change  by  the  amount  represented  by  the  distance 
between  any  two  of  these  divisions.  This  distance  is  found  to  be  *  of 
the  height  ED. 


Fi«. 124.  The  stand- 
ard gas  thermometer 


THERMOMETEY  121 

In  other  words,  one  degree  of  change  in  temperature  is  such 
a  temperature  change  as  will  cause  the  pressure  exerted  by  a 
confined  gas  to  change  by  -^j  °f  its  value  at  the  temperature  of 
melting  ice  (0°  C.). 

159.  Absolute  temperature.    Since,  then,  cooling  the  hydro- 
gen through  1°  C.,  as  defined  above,  reduces  the  pressure  -^j 
of  its  value  at  0°  C.,  it  is  clear  that  cooling  it  273°  below  0°  C. 
would  reduce  its  pressure  to  nothing.    But  from  the  stand- 
point of  the  kinetic  theory  this  would  be  the  temperature  at 
which  all  motions  of  the  hydrogen  molecules  would  cease. 
This  temperature  is  called  the  absolute  zero  and  the  temper- 
ature measured  from  this  zero  is  called  absolute  temperature. 
Thus,  if  A   is  used  to  denote   the  absolute  scale,  we  have 
0°  C.  =  273°  A.,  100°  C.  =  373°  A.,  15°  C.  =  288°  A.,  etc.    It 
is  customary  to  indicate  temperatures  on  the  centigrade  scale 
by  £,  on  the  absolute  scale  by  T.    We  have  then 

T=£+273.  (1) 

160.  Comparison  of  gas  and  mercury  thermometers.    Since  an  inter- 
national committee  has  chosen  the  hydrogen  thermometer  described  itt 
§  158  as  the  standard  of  temperature  measurement,  it  is  important  to 
know  whether  mercury  thermometers,  graduated  in  the  manner  described 
in  §  154,  agree  with  gas  thermometers  at  temperatures  other  than  0°  and 
100°,  where,  of  course,  they  must  agree,  since  these  temperatures  are  in 
each  case  the  starting  points  of  the  graduation.    A  careful  comparison 
has  shown  that  although  they  do  not  agree  exactly,  yet  fortunately  the 
disagreements  at  ordinary  temperatures  are  small,  not  amounting  to  more 
than  .2°  anywhere  between  0°  and  100°.    At  300°  C.,  however,  the  differ- 
ence amounts  to  about  4°.    (Mercury  thermometers  are  actually  used  for 
measuring  temperatures  above  the  boiling  point  of  mercury,  360°  C.  They 
are  then  filled  with  nitrogen,  the  pressure  of  which  prevents  boiling.) 

Hence  for  all  ordinary  purposes  mercury  thermometers  are  sufficiently 
accurate,  and  no  special  standardization  of  them  is  necessary.  But  in  all 
scientific  work,  if  mercury  thermometers  are  used  at  all,  they  must  first 
be  compared  with  a  gas  thermometer  and  a  table  of  corrections  obtained. 
The  errors  of  an  alcohol  thermometer  are  considerably  larger  than  those 
of  a  mercury  thermometer. 


122    THEEMOMETEY;  EXPANSION  COEFFICIENTS 


161.  Low  temperatures.    The  absolute  zero  of  temperature 
can,  of  course,  never  be  attained,  but  in  recent  years  rapid 
strides  have  been  made  toward  it.    Forty  years  ago  the  low- 
est temperature  which  had  ever  been  measured  was  —110°  C., 
the  temperature  attained  by  Faraday  in  1845  by  causing  a 
mixture  of  ether  and  solid  carbon  dioxide  to  evaporate  in 
a  vacuum.    But  in  1880  air  was  first  liquefied,  and  found, 
by  means  of  a  gas  thermometer,  to  have  a  temperature  of 

- 180°  C.  When  liquid  air  evaporates  into  a  space  which 
is  kept  exhausted  by  means  of  an  air  pump,  its  temperature 
falls  to  about  —  220°  C.  Recently  hydrogen  has  been  lique- 
fied and  found  to  have  a  temperature  at  atmospheric  pressure 
of  —  243°  C.  All  of  these  temperatures  have  been  measured 
by  means  of  hydrogen  thermometers.  By 
allowing  liquid  hydrogen  to  evaporate  into 
a  space  kept  exhausted  by  an  air  pump, 
Dewar  in  1900  attained  a  temperature 
of  -  260°.  In  1911  Kamerlingh  Onnes 
liquefied  helium  and  attained  a  tempera- 
ture of  -  271.3°  C.,  only  1.7°  above  abso- 
lute zero  (see  §  92). 

162.  Maximum  and  minimum  thermometers.   In 

all  weather  bureaus  the  lowest  temperature  reached 
during  the  night,  and  the  highest  temperature 
reached  during  the  day,  are  automatically  recorded 
by  a  special  device  called  a  maximum  and  mini- 
mum thermometer.  The  construction  of  one  form 
of  this  instrument  is  shown  in  Fig.  125.  The 
bulb  A  and  the  stem  down  to  the  point  G  are 
filled  with  alcohol,  from  G  to  B  the  stem  is  filled 
with  mercury,  while  the  liquid  above  B  is  again 
alcohol.  The  bulb  D  contains  only  alcohol  and 

its  vapor.  The  two  indices  d  and  C  move  with  slight  friction  in  the 
stem.  As  the  temperature  falls,  the  alcohol  in  A  contracts  and  the  mer- 
cury pushes  up  the  index  on  the  right  and  leaves  it  opposite  the  mark 
corresponding  to  the  lowest  temperature  reached.  As  the  temperature 


"Qt 

85- 

-20 
-15 

80- 

-10 

75. 

-5 

70- 

-0 

65- 

-5 

*0- 

-10 

55- 

-C 

50- 

-20 

45- 

-25 

40- 

-30 

35- 

-35 

at- 

-40 

25- 

-45 

20- 

-50 

15- 

-55 

10- 

-60 

5- 

-65 

o-     ^ 

P        -70 

5- 

!°[i    a 
% 

-75 

gr_ 

FIG.  125.    The  maxi- 
mum and  minimum 
thermometer 


SIR  WILLIAM  THOMSON,  LORD  KELVIN  (1824-1907) 

One  of  the  best  known  and  most  prolific  of  nineteenth-century  physicists ;  born 
in  Belfast,  Ireland ;  professor  of  physics  in  Glasgow  University,  Scotland,  for 
more  than  fifty  years;  especially  renowned  for  his  investigations  in  heat  and 
electricity;  originator  of  the  absolute  thermodynamic  scale  of  temperature; 
formulator  of  the  second  law  of  thermodynamics ;  inventor  of  the  electrometer, 
the  mirror  galvanometer,  and  many  other  important  electrical  devices 


EXPANSION  COEFFICIENTS  123 

rises,  the  alcohol  in  A  expands  and  the  mercury  pushes  up  the  index 
on  the  left  and  leaves  it  opposite  the  mark  corresponding  to  the 
highest  temperature  reached.  In  order  to  obtain  the  right  amount  of 
friction,  a  small  steel  spring  is  attached  to  the  indices,  as  in  A'.  After 
each  observation  the  observer  pulls  the  index  back  to  contact  with  the 
mercury  by  means  of  a  small  magnet. 

QUESTIONS  AND  PROBLEMS 

1.  Normal  room  temperature  is  68°  F.    What  is  it  centigrade? 

2.  The  normal  temperature  of  the  human  body  is  98.4°  F.  What  is 
it  centigrade  ? 

3.  What  temperature  centigrade  corresponds  to  0°  F.  ? 

4.  Mercury  freezes  at  —  40°  F.   What  is  this  centigrade  ? 

5.  The  temperature  of  liquid  air  is  —  180°  C.  What  is  it  Fahrenheit  ? 

6.  The  lowest  temperature  attainable  by  evaporating  liquid  helium 
is  -  271.4°  C.    What  is  it  Fahrenheit? 

7.  What  is  the  absolute  zero  of  temperature  on  the  Fahrenheit  scale  ? 

8.  Why  is  a  fever  thermometer  made  with  a  very  long  cylindrical 
bulb,  instead  of  a  spherical  one  ? 

9.  When  the  bulb  of  a  thermometer  is  placed  in  hot  water,  it  at 
first  falls  a  trifle  and  then  rises.    Why? 

10.  How  does  the  distance  between  the  0°  mark  and  the  100°  mark 
vary  with  the  size  of  the  bore,  the  size  of  the  bulb  remaining  the  same  ? 

11.  What  is  meant  by  the  absolute  zero  of  temperature? 

12.  Why  is  the  temperature  of  liquid  air  lowered  if  it  is  placed  under 
the  receiver  of  an  air  pump  and  the  air  exhausted  ? 

13.  Two  thermometers  have  bulbs  of  equal  'size.    The  bore  of  one 
has  a  diameter  twice  that  of  the  other.    What  are  the  relative  lengths 
of  the  stems  between  0°  and  100°  ? 


EXPANSION  COEFFICIENTS 

163.  The  laws  of  Charles  and  Gay-Lussac.  When,  as  in  the 
experiment  described  in  §  158,  we  keep  the  volume  of  a  gas 
constant  and  observe  the  rate  at  which  the  pressure  increases 
with  rise  in  temperature,  we  obtain  the  pressure  coefficient  of 
expansion,  which  is  defined  as  the  ratio  between  the  increase  in 
pressure  per  degree  and  the  value  of  the  pressure  at  0°  C.  This 
was  first  done  for  different  gases  by  a  Frenchman,  Charles, 


124     THEKMOMETKY;   EXPANSION  COEFFICIENTS 

in  1787,  who  found  that  the  pressure  coefficients  of  expansion  of 
all  gases  are  the  same.  This  is  known  as  the  law  of  Charles. 

When  we  arrange  the  experiment  so  that  the  gas  can  expand 
as  the  temperature  rises,  the  pressure  remaining  constant,  we 
obtain  the  volume  coefficient  of  expansion,  which  is  defined  as  the 
ratio  between  the  increase  in  volume  per  degree  and  the  total  vol- 
ume of  the  gas  at  0°  (7.  This  was  first  done  for  different  gases  in 
1802  by  another  Frenchman,  Gay-Lussac,  who  found  that  all 
gases  have  the  same  volume  coefficient  of  expansion,  this  coefficient 
being  the  same  as  the  pressure  coefficient,  namely  1/273.  This 
is  known  as  the  law  of  G-ay-Lussac. 

From  the  definition  of  absolute  temperature  and  Charles's 
law  we  learn  that  for  all  gases  at  constant  volume,  pressure  is 
proportional  to  absolute  temperature  ;  that  is, 

P         T 


Also  from  Gay-Lussac's  law  we  learn  that  for  all  gases  at 
constant  pressure,  volume  is  proportional  to  absolute  temperature; 
that  is, 

-l-  =  i>.  (3) 

V-i      T* 

If  pressure,  temperature,  and  volume  all  vary,*  we  have 
P  V        T 

±  1  Y  \  __  ±\  (A*\ 

P*V*~   T* 

Any  one  of  these  six  quantities  may  be  found  if  the  other 
five  are  known. 

If  the  volume  remains  constant,  that  is,  if  F1  =  V^  equation 
(4)  reduces  to  (2)  ;  that  is,  to  Charles's  law.  If  the  pressure 

*  If  this  is  not  clear  to  the  student,  let  him  recall  that  if  the  speeds  of  two 
runners  are  the  same,  then  their  distances  are  proportional  to  their  times, 
that  is,  D-L/DZ  =  ti/tz  ;  hut  if  their  times  are  the  same  and  the  speeds  different, 
D^/Dz  =s1/«2.  If  now  one  runs  hoth  twice  as  fast  and  twice  as  long,  he  evidently 
goes  4  times  as  far,  that  is,  if  time  and  speed  hoth  vary,  D^/Dz 


EXPANSION  OF  LIQUIDS  AND  SOLIDS          125 

remains  constant,  P1  =  P0  and  equation  (4)  reduces  to  (8);  that 
is,  to  Gay-Lussac's  law.  If  the  temperature  does  not  change, 
Tl  =  T2  and  equation  (4)  reduces  to  P1  Vl  =  Po  F2 ;  that  is,  to 
Boyle's  law.  If  the  relation  of  densities  instead  of  volumes  are 

V  I) 

sought,  it  is  only  necessary  to  replace  —  in  (3)  and  (4)  by  — -. 

QUESTIONS  AND  PROBLEMS 

1.  What  fractional  part  of  the  air  in  a  room  passes  out  when  the  air  in 
it  is  heated  from  -  15°C.  to  20°C.?  (-  15°C.  =  258° A.;  20°  C.  =  293° A.) 

2.  Why  is  it  unsafe  to  let  a  pneumatic  inkstand  like  that  of  Fig.  32, 
p.  32,  remain  in  the  sun?  What  changes  will  occur  in  the  volume  of 
the  confined  gas  if  it  is  heated,  from  15°  C.  to  40°  C.  ? 

3.  To  what  temperature  must  a  cubic  foot  of  gas  initially  at  0°  C. 
be  raised  in  order  to  double  its  volume,  the  pressure  remaining  constant? 

•*•  4.  If  the  air  within  a  bicycle  tire  is  under  a  pressure  of  2  atmospheres, 
that  is,  152  cm.  of  mercury,  when  the  temperature  is  10°  C.,  what  pressure 
will  exist  within  the  tube  when  the  temperature  changes  to  35°  C.  ? 

*-  5.  If  the  pressure  to  which  15  cc.  of  air  is  subjected  changes  from 
76  cm.  to  40  cm.,  the  temperature  remaining  constant,  what  does  its 
volume  become  ?  (See  Boyle's  law,  p.  35.)  If,  then,  the  temperature  of 
the  same  gas  changes  from  15°  C.  to  100°  C.,  the  pressure  remaining 
constant,  what  will  be  the  final  volume? 

*^  6.  If  the  volume  of  a  gas  at  20°  C.  and  76  cm.  pressure  is  500  cc., 
what  is  its  volume  at  50°  C.  and  70  cm.  pressure? 


EXPANSION  OF  LIQUIDS  AND  SOLIDS 

164.   The  expansion  of  liquids.     The  expansion  of  liquids 
differs  from  that  of  gases  in  that 

1.  The   coefficients   of   expansion   of    liquids   are   all   con- 
siderably smaller  than  those  of  gases. 

2.  Different  liquids  expand  at  wholly  different  rates;  for 
example,  the  coefficient  of  alcohol  between  0°  and  10°  C.  is 
.0011 ;  of  ether  it  is  .0015  ;  of  petroleum,  .0009. 

3.  The  same  liquid  often  has  different  coefficients  at  differ- 
ent temperatures;  that  is,  the  expansion  is  irregular.    Thus, 


126     THEEMOMETEY;   EXPANSION  COEFFICIENTS 


if  the  coefficient  of  alcohol  is  obtained  between  0°  and  60°  C., 
instead  of  between  0°  and  10°  C.,  it  is  .0018  instead  of  .0011. 

The  coefficient  of  mercury,  however,  is  very 
nearly  constant  through  a  wide  range  of  temper- 
ature, which  indeed  might  have  been  inferred 
from  the  fact  that  mercury  thermometers  agree 
so  well  with  gas  thermometers. 

165.  Method  of  measuring  the  expansion  coeffi- 
cients of  liquids.    One  of  the  most  convenient 
and  common  methods  of  measuring  the  coeffi- 
cients of  liquids  is  to  place  them  in  bulbs  of 
known  volume,  provided  with  capillary  necks 
of  known  diameter,  like  that  shown  in  Fig.  126, 
and  then  to  watch  the  rise  of  the  liquid  in  the 
neck  for  a  given  rise  in  temperature.   A  certain 
allowance  must  be  made  for  the  expansion  of  the 

bulb,  but  this  can  readily  be  done  if  the  coefficient  of  expan- 
sion of  the  substance  of  which  the  bulb  is  made  is  known. 

166.  Maximum  density  of  water.  When 
water  is  treated  in  the  way  described  in 
the  preceding  paragraph,  it  reaches  its 
lowest  position  in  the  stem  at  4°  C.    As 
the  temperature  falls   from  that  point 
down  to  0°  C.,  water  exhibits  the  pecu- 
liar property  of  expanding  with  a  decrease 
in  temperature. 


FIG.  126.  Bulb 
or  investiat- 


FIG.  127.  Maximum 
density  of  water 


This  may  be  shown  experimentally  by  sur- 
rounding for  an  hour  or  more  a  vessel  of  water 
with  a  freezing  mixture  of  ice  and  salt  as  in 
Fig.  127.  The  lower  thermometer  will  first  fall 
to  4°C.  and  remain  there,  after  which  the  upper  thermometer  will  fall  rap- 
idly, showing  that  water  colder  than  4°  C.  is  now  rising  instead  of  falling. 

We  learn,  therefore,  that  water  has  its  maximum  density  at 
a  temperature  of  4°  C. 


EXPANSION  OF  LIQUIDS  AND  SOLIDS          127 

167.  The  cooling  of  a  lake  in  winter.    The  preceding  para- 
graph makes  it  easy  to  understand  the  cooling  of  any  large 
body  of  water  with  the  approach  of  winter.    The  surface  layers 
are  first  cooled  and  contract.    Hence,  being  then  heavier  than 
the  lower  layers,  they  sink  and  are  replaced  by  the  warmer 
water  from  beneath.    This  process  of  cooling  at  the  surface, 
and  sinking,  goes  on  until  the  whole  body  of  water  has  reached 
a  temperature  of  4°  C.    After  this  condition  has  been  reached, 
further  cooling  of  the  surface  layers  makes  them  lighter  than 
the  water  beneath,  and  they  now  remain  on  top  until  they 
freeze.   Thus,  before  any  ice  whatever  can  form  on  the  surface 
of  a  lake,  the  whole  mass  of  water  to  the  very  bottom  must 
be  cooled  to  4°  C.    This  is  why  it  requires  a  much  longer  and 
more  severe  period  of  cold  to  freeze  deep  bodies  of  water  than 
shallow  ones.    Further,  since  the  circulation  described  above 
ceases  at  4°  C.,  practically  all  of  the  unfrozen  water  will  be 
at  4°  C.  even  in  the  coldest  weather.    Only  the  water  which  is 
in  the  immediate  neighborhood  of  the  ice  will  be  lower  than 
4°  C.    This  fact  is  of  vital  importance  in  the  preservation  of 
aquatic  life. 

168.  Linear  coefficients  of  expansion  of  solids.    It  is  often 
more   convenient  to   measure  the  increase   in  length  of  one 
edge  of  an  expanding  solid  than  to  measure  its  increase  in 
volume.    The  ratio  between  the  increase  in  length  per  degree  rise 
in  temperature  and  the  total  length  is  called  the  linear  coeffi- 
cient of  expansion  of  the  solid.    Thus,  if  ^  represent  the  length 
of  a  bar  at  ^°,  and  lz  its  length  at  £0°,  the  equation  which 
defines  the  linear  coefficient  k  is 

k=     *2~Zl (5) 

u*,-*!) 

Fig.  128  illustrates  the  method  now  in  use  at  the  Interna- 
tional Bureau  of  Weights  and  Measures  for  obtaining  these 
coefficients.  The  two  microscopes  which  are  mounted  in  fixed 


128     THEEMOMETRY ;   EXPANSION  COEFFICIENTS 


Microscope 


Glass  .  . 

.  .000009 

Silver  . 

.  .000019 

Iron  .  . 

.  .000012 

Steel  .  . 

.  .000011 

Lead  .  . 

.  .000029 

Tin  .  . 

.  .000022 

Platinum 

.000009 

Zinc  . 

.000029 

positions  upon  heavy  piers  are  focused  upon  scratches  near 
the  ends  of  the  bar  whose  coefficient  is  to  be  obtained.  The 
temperature  of  the  water  is 
then  changed  from,  say,0°C. 
to  10°  C.,  and  the  amount  of 
elongation  of  the  bar  is  de- 
termined from  the  observed 
amounts  of  motion  of  its 
ends  as  seen  through  the 
microscopes. 

The  linear  coefficients  of  a 
few  common  substances  are    FlG.  128.   Apparatus  for  determination 
given  ill  the  following  table  :          of  linear  coefficients  of  expansion 

Aluminium  .000023 

Brass      .    .  .000018 

Copper  .    .  .000017 

Gold  .    .    .  .000014 

APPLICATIONS  OF  EXPANSION 

169.  Compensated  pendulum.  Since  a  long  pendulum  vibrates 
more  slowly  than  a  short  one,  the  expansion  of  the  rod  which 
carries  the  pendulum  bob  causes  an  ordinary 
clock  to  run  too  slowly  in  summer,  and  its 
contraction  causes  it  to  run  too  fast  in  winter. 
For  this  reason  very  accurate  clocks  are  pro- 
vided with  compensated  pendulums,  which  are 
so  constructed  that  the  distance  of  the  bob 
beneath  the  point  of  support  is  independent 
of  the  temperature.  This  is  accomplished  by 
suspending  the  bob  by  means  of  two  sets 
of  rods  of  different  material,  in  such  a  way 
that  the  expansion  of  one  set  raises  the  bob, 
while  the  expansion  of  the  other  set  lowers  FIG  ^phe  corn- 
it.  Such  a  pendulum  is  shown  in  Fig.  129.  pensated  pendulum 


APPLICATIONS  OF  EXPANSION 


129 


The  expansion  of  the  iron  rods  £>,  d,  e,  and  i  tends  to  lower 
the  bob,  while  that  of  the  copper  rods  c  tends  to  raise  it.  In 
order  to  produce  complete  compensation  it  is 
only  necessary  to  make  the  total  lengths  of  iron 
and  copper  rods  inversely  proportional  to  the 
coefficients  of  expansion  of  iron  and  copper. 

170.  Compensated  balance  wheel.    In  the  bal- 
ance wheel  of  an  accurate  watch  (Fig.  130)  an-    FIG.  130.  The 
.-,  -,.     ,.  «    .*,  T  .  £     compensated 

other  application   of  the  unequal  expansion  ot    balance  Wheel 

metals  is  made.    Increase  in  temperature  both 

increases  the  radius  of  the  wheel  and  weakens  the  elasticity 

of  the  spring  which  controls  it.    Both  of  these  effects  tend  to 


FIG.  131  FIG.  132 

Unequal  expansion  of  metals 

make  the  watch  lose  time.  This  tendency  may  be  counteracted 
by  bringing  the  mass  of  the  rotating  parts  in  toward  the  center 
of  the  wheel.  This  is  accomplished  by  making  the  arcs  be  of 
metals  of  different  expansion  coefficients,  the 
inner  metal,  shown  in  black  in  the  figure,  having 
the  smaller  coefficient.  The  weighted  ends  of  the 
arcs  are  then  sufficiently  pulled  in  by  a  rtee  in 
temperature  to  counteract  the  retarding  effects. 

The  principle  is  precisely  the  same  as  that  which 
finds  simple  illustration  in  the  compound  bar  shown  in 
Fig.  131.  This  bar  consists  of  two  strips,  one  of  brass 
and  one  of  iron,  riveted  together.  When  the  bar  is 
placed  edgewise  in  a  Bunsen  flame,  so  that  both  metals 
are  heated  equally,  it  will  be  found  to^bend  in  such  a 
way  that  the  more  expansible  metal,  namely  the  brass, 
is  on  the  outside  of  the  curve,  as  shown  in  Fig.  132. 
When  it  is  cooled  with  snow  or  ice  it  bends  in  the 
opposite  direction. 

The  common  thermostat  (Fig.  133)  is  precisely  such 
a  bar  which  is  arranged  so  as  to  open  the  drafts 

t 


133.   The 
thermostat 


130     THEEMOMETRY;   EXPANSION  COEFFICIENTS 


through  closing  an  electrical  circuit  at  a  when  it  is  too  cold,  and 
to  close  the  drafts  through  making  contact  at  b  when  it  is  too  warm. 

171.  The  dial  thermometer.  The  dial  thermometer 
is  a  compound  metallic  ribbon  wound  in  helical  form. 
One  end  a  of  the  helix  [Fig.  134,  (2)]  is  fixed,  while 


(1) 


FIG.  134.   The  dial  thermometer 


the  other  end  is  attached 
to  a  lever  arm  b,  the  mo- 
tion of  which  rotates  the 
pointer  d  over  the  dial 
[Fig.  134,  (1)],  which  is 
graduated  by  comparison 
with  a  mercury  thermom- 
eter. The  more  expansible 
metal  is  on  the  outside. 
Hence  rise  in  temperature 
causes  the  helix  to  wind 
up  closer,  the  index  mov- 
ing to  the  right. 

QUESTIONS  AND  PROBLEMS 

1.  What  is  the  temperature  of  the  water  at  the  bottom  of  a  lake  in  very 
cold  weather?  Test  the  temperature  of  your  tap  water  in  winter.  Explain. 

2.  Why  is  a  thick  tumbler  more  likely  to  break  when  hot 
water  is  poured  into  it  than  a  thin  one  ? 

3.  Why  may  a  glass  stopper  sometimes  be  loosened  by 
pouring  hot  water  on  the  neck  of  a  bottle  ? 

4.  If  an  iron  steam  pipe  is  60  ft.  long  at  0°C.,  what  is  its 
length  when  steam  passes  through  it  at  100° C.? 

5.  Pendulums  are  often  compensated  by  using  cylinders  of 
mercury  as  in  Fig.  135.    Explain. 

6.  The  steel  cable  from  which  Brooklyn  Bridge  hangs  is 
more  than  a  mile  long.    By  how  many  feet  does  a  mile  of 
its  length  vary  between  a  winter  day  when  the  temperature 
is  —  20°  C.,  and  a  summer  day  when  it  is  30°  C.  ? 

7.  The  changes  in  temperature  to  which   long  lines  of 
steam  pipes  are  subjected  make  it  necessary  to  introduce 
"expansion  joints."  These  joints  consist  of  brass  collars  fitted 
tightly  by  means  of  packing  over  the  separated  ends  of  two 
adjacent  lengths  of  pipe.    If  the  pipe  is  of  iron,  and  such  a    FIG.  135 
joint  is  inserted  every  200  ft.,  and  if  the  range  of  tempera- 
ture which  must  be  allowed  for  is  from  —  30°  C.  to  125°  C.,  what  is  the 
minimum  play  which  must  be  allowed  for  at  each  expansion  joint  ? 


CHAPTER 

WORK  AND  MECHANICAL  ENERGY* 
DEFINITION  AND  MEASUREMENT  OF  WOKK 

172.  Definition  of  work.  Whenever  a  force  moves  a  body  on 
which  it  acts,  it  is  said  to  do  work  upon  that  body;  and  the 
amount  of  the  work  accomplished  is  measured  by  the  product 
of  the  force  acting  and  the  distance  through  which  it  moves  the 
body.  Thus,  if  1  gram  of  mass  is  lifted  1  centimeter  in  a  verti- 
cal direction,  1  gram  of  force  has  acted,  and  the  distance  through 
which  it  has  moved  the  body  is  1  centimeter.  We  say,  there- 
fore, that  the  lifting  force  has  accomplished  1  gram  centimeter 
of  work.  If  the  gram  of  force  had  lifted  the  body  upon  which 
it  acted  through  2  centimeters,  the  work  done  would  have  been 
2  gram  centimeters.  If  a  force  of  3  grams  had  acted  and  the 
body  had  been  lifted  through  3  centimeters,  the  work  done 
would  have  been  9  gram  centimeters,  etc.  Or,  in  general,  if 
W  represent  the  work  accomplished,  F  the  value  of  the  acting 
force,  and  s  the  distance  through  which  its  point  of  application 
moves,  then  the  definition  of  work  is  given  by  the  equation 

W=Fxs.  (1) 

In  the  scientific  sense  no  work  is  ever  done  unless  the  force 
succeeds  in  producing  motion  in  the  body  on  which  it  acts.  A 
pillar  supporting  a  building  does  no  work ;  a  man  tugging  at 
a  stone,  but  failing  to  move  it,  does  no  work.  In  the  popular 

*  It  is  recommended  that  this  chapter  be  preceded  by  an  experiment  in  which 
the  student  discovers  for  himself  the  law  of  the  lever,  that  is,  the  principle  of 
moments  (see,  for  example,  Experiment  16,  authors'  manual),  and  that  it  be 
accompanied  by  a  study  of  the  principle  of  work  as  exemplified  in  at  least  one  of 
the  other  simple  machines  (see,  for  example,  Experiment  17,  authors'  manual). 

131 


132     WOEK  AND  MECHANICAL  ENERGY 

sense  we  sometimes  say  that  we  are  doing  work  when  we  are 
simply  holding  a  weight,  or  doing  anything  else  which  results 
in  fatigue  ;  but  in  physics  the  word  "  work  "  is  used  to  describe 
not  the  effort  put  forth  but  the  effect  accomplished,  as  repre- 
sented in  equation  (1). 

173.  Units  of  work.  There  are  two  common  units  of  work 
in  the  metric  system,  the  gram  centimeter  and  the  kilogram 
meter.  As  the  names  imply,  the  gram  centimeter  is  the  work 
done  by  a  force  of  1  gram  when  it  moves  the  point  on  which  it 
acts  1  centimeter.  The  kilogram  meter  is  the  work  done  by  a 
kilogram  of  force  when  it  moves  the  point  on  which  it  acts 
1  meter.  The  gram  meter  is  also  sometimes  used. 

Corresponding  to  the  English  unit  of  force,  the  pound,  is  the 
unit  of  work,  the  foot  pound.  It  is  the  work  done  by  a  "  pound  of 
force  "  when  it  moves  the  point  on  which  it  acts  1  foot.  Thus 
it  takes  a  foot  pound  of  work  to  lift  a  pound  of  mass  1  foot  high. 

In  the  absolute  system  of  units  the  dyne  is  the  unit  of  force  and  the 
dyne  centimeter,  or  erg,  is  the  corresponding  unit  of  work.  The  erg  is 
the  amount  of  work  done  by  a  force  of  1  dyne  when  it  moves  the  point 
on  which  it  acts  1  centimeter.  To  raise  1  liter  of  water  from  the  floor 
to  a  table  1  meter  high  would  require  1000  x  980  x  100  =  98,000,000  ergs 
of  work.  It  will  be  seen,  therefore,  that  the  erg  is  an  exceedingly  small 
unit.  For  this  reason  it  is  customary  to  employ  a  unit  which  is  equal 
to  10,000,000  ergs.  It  is  called  a  joule,  in  honor  of  the  great  English 
physicist  James  Prescott  Joule  (1818-1889).  The  work  done  in  lifting 
a  liter  of  water  1  meter  is  therefore  9.8  joules. 

QUESTIONS  AND  PROBLEMS 

1.  Analyze  several  types  of  manual  labor  and  see  if  the  above  defini- 
tion (  W=  Fs)  holds  for  each.  Is  not  F  x  s  the  thing  paid  for  in  every  case  ? 

2.  How  many  foot  pounds  of  work  does  a  150-lb.  man  do  in  climbing 
to  the  top  of  Mt.  Washington,  which  is  6300  ft.  high  ? 

3.  A  horse  pulls  a  metric  ton  of  coal  to  the  top  of  a  hill  30  m.  high. 
Express  the  work  accomplished,  first  in  kg.  m.,  then  in  joules. 

4.  If  the  20,000  inhabitants  of  a  city  use  an  average  of  20  1.  of  water 
per  day  per  capita,  how  many  kilogram  meters  of  work  must  the  engines 
do  per  day,  if  the  water  has  to  be  raised  to  a  height  of  75  m.  ? 


WOEK  AND  THE  PULLEY 


133 


FIG.  136.  The 

single      fixed 

pulley 


WOEK  EXPENDED  UPON  AND  ACCOMPLISHED  BY  SYSTEMS 
OP  PULLEYS 

174.  The  Single  fixed  pulley.    Let  the  force  of  the  earth's  attrac- 
tion upon  a  mass  F'  'be  overcome  by  pulling  upon  a  spring  balance  S, 
in  the  manner  shown  in  Fig.  136,  until  F'  moves  slowly 

upward.  If  F'  is  100  grains,  the  spring  balance  will  also 
be  found  to  register  a  force  of  100  grams. 

Experiment  therefore  shows  that  in  the  use 
of  the  single  fixed  pulley  the  acting  force  F 
which  is  producing  the  motion  is  equal  to  the 
resisting  force  F'  which  is  opposing  the  motion. 

Again,  since  the  length  of  the  string  is  always 
constant,  the  distance  s  through  which  the  point 
A,  at  which  F  is  applied,  must  move,  is  always 
equal  to  the  distance  sf  through  which  the  weight 
F'  is  lifted.  Hence,  if  we  consider  the  work  put 
into  the  system  at  A,  namely  F  x  s,  and  the  work  accomplished 
by  the  system  at  Ff,  namely  F'  x  s',  we  find  obviously,  since 
F'  =  F  and  s  =  »',  that 

Fs  =  F'sf-,  (2) 

that  is,  in  the  case  of  the  single  fixed  pulley,  the 
work  done  by  the  acting  force  F  (the  effort*)  is 
equal  to  the  work  done  against  the  resisting  force 
F'  (the  resistance)  ;  or  the  work  put  into  the  ma- 
chine at  A  is  equal  to  the  work  accomplished  by 
the  machine  at  F'. 

175.  The  single  movable  pulley.    Let  now  the 

force  of  the  earth's  attraction  upon  the  mass  F'  be  over- 
come by  a  single  movable  pulley,  as  shown  in  Fig.  137.  ^IG>  137<  The 
Since  the  weight  of  F'  (Ff  representing  in  this  case  the  single  movable 
weight  of  both  the  pulley  and  the  suspended  mass)  is 
now  supported  half  by  the  strand  C  and  half  by  the  strand  B,  the 
force  F  which  must  act  at  A  to  hold  the  weight  in  place,  or  to  move 
it  slowly  upward  if  there  is  no  friction,  should  be  only  one  half  of  F'. 
A  reading  of  the  balance  will  show  that  this  is  indeed  the  case. 


134 


WORK  AND  MECHANICAL  ENERGY 


Experiment  thus  shows  that  in  the  case  of  the  single 
movable  pulley  the  effort  F  is  just  one  half  as  great  as  the 
resistance  F'. 

But  when  again  we  consider  the  work  which  the  force  F 
must  do  to  lift  the  weight  F'  a  distance  «',  we  see  that  A  must 
move  upward  2  inches  in  order  to  raise  F1  1  inch.  For  when 
F1  moves  up  1  inch  both  of  the  strands  B  and  C  must  be 
shortened  1  inch.  As  before,  therefore,  since  F'  =  2  F,  and 


that  is,  in  the  case  of  the  single  movable  pulley,  as  in  the 
case  of  the  fixed  pulley,  the  work  put  into  the  machine  by  the 
effort  F  is  equal  to  the  work  accom- 
plished by  the  machine  against  the 
resistance  F'. 


(l) 


176.  Combinations  of  pulleys.  Let 

a  weight  F'  be  lifted  by  means  of  such  a 
system  of  pulleys  as  is  shown  in  Fig.  138, 
either  (1)  or  (2).  Here,  since  F'  is  sup- 
ported by  6  strands  of  the  cord,  it  is  clear 
that  the  force  which  must  be  applied  at  A 
in  order  to  hold  F'  in  place,  or  to  make  it 
move  slowly  upward  if  there  is  no  friction, 
should  be"  but  I  of  F'. 


FIG.  138. 


Combinations   of 
pulleys 


The  experiment  will  show  this  to 
be  the  case  if  the  effects  of  friction, 
which  are  often  very  considerable, 
are  eliminated  by  taking  the  mean 
of  the  forces  which  must  be   ap- 
plied at  F  to  cause  it  to  move  first  slowly  upward  and  then 
slowly   downward.     The  law   of    any  combination    of   mov- 
able  pulleys   may  then   be   stated   thus:   If  n  represent  the 
number  of  strands  between  which  the  iveight  is  divided, 

F  =  F'/n. 


WOKK  AND  THE  PULLEY  135 

But  when  again  we  consider  the  work  which  the  force  F 
must  do  in  order  to  lift  the  weight  F'  through  a  distance  s', 
we  see  that,  in  order  that  the  weight  F1  may  be  moved  up 
through  1  inch,  each  of  the  strands  must  be  shortened  1  inch, 
and  hence  the  point  A  must  move  through  n  inches ;  that  is, 
s'  —  s/n.  Hence,  ignoring  friction,  in  this  case  also  we  have 

F  x  s  =  Ff  x  «' ; 

that  is,  although  the  effort  F  is  only  -  of  the  resistance  F\  the 

work  put  into  the  machine  by  the  effort  F  is  equal  to  the  work 
accomplished  by  the  machine  against  the  resistance  F'. 

177.  Mechanical  advantage.  The  above  experiments  show 
that  it  is  sometimes  possible,  by  applying  a  small  force  F, 
to  overcome  a  much  larger  resisting  .force  F'.  The  ratio 
of  the  resistance  F'  to  the  effort  F  is  called  the  mechanical 
advantage  of  the  machine.  Thus  the  mechanical  advantage 
of  the  single  fixed  pulley  is  1,  that  of  the  single  movable 
pulley  is  2,  that  of  the  systems  of  pulleys  shown  in  Fig.  138 
is  6,  etc. 

If  the  acting  force  is  applied  at  F'  instead  of  at  F,  the  me- 
chanical advantage  of  the  systems  of  pullers  of  Fig.  138  is  ^ ; 
for  it  requires  an  application  of  6  pounds  at  Fr  to  lift  1  pound 
at  F.  But  it  will  be  observed  that  the  resisting  force  at  F  now 
moves  six  times  as  fast  and  six  times  as  far  as  the  acting  force 
at  F'.  We  can  thus  either  sacrifice  speed  to  gain  force,  or 
sacrifice  force  to  gain  speed ;  but  in  every  case,  whatever  we 
gain  in  the  one  we  lose  in  the  other.  Thus  in  the  hydraulic 
elevator  shown  in  Fig.  13,  p.  18,  the  cage  moves  only  as  fast 
as  the  piston  ;  but  in  that  shown  in  Fig.  14  it  moves  four  times 
as  fast.  Hence  the  force  applied  to  the  piston  in  the  latter 
case  must  be  four  times  as  great  as  in  the  former  if  the  same 
load  is  to  be  lifted.  This  means  that  the  diameter  of  the  latter 
cylinder  must  be  twice  as  great. 


136  WOEK  AND  MECHANICAL  ENEKGY 

QUESTIONS  AND  PROBLEMS 

1.  Since  the  mechanical  advantage  of  a  single  fixed  pulley  is  only  1, 
why  is  it  ever  used? 

2.  If  the  hydraulic  elevator  of  Fig.  14,  p.  18,  is  to  carry  a  total  load 
of  20,000  lb.,  what  force  must  be  applied  to  the  piston?   If  the  water 
pressure  is  70  lb.  per  square  inch,  what  must  be  the  area  of  the  cross 
section  of  the  piston? 

3.  Draw  a  diagram  of  a  set  of  pulleys  by  which  a  force  of  50  lb.  can 
support  a  load  of  200  lb. 

4.  Draw  a  diagram  of  a  set  of  pulleys  by  which  a  force  of  50  lb.  can 
support  250  lb.    What  would  be  the  mechanical  advantage   of   this 
arrangement  ? 

WORK  AND  THE  LEVEE, 

V 

178.  The  law  of  the  lever.    The  lever  is  a  rigid  rod  free1!o\ 
turn  about  some  point  P  called  the  fulcrum  (Fig.  139). 

Let  a  meter  stick  be  first  balanced  as  in  the  figure,  and  then  let  a 
mass,  of,  say,  300  g.,  be  hung  by  a  thread  from  a  point  15  cm.  from 
the  fulcrum.  Then 
let  a  point  be  found 
on  the  other  side  of 
the  fulcrum  at  which 
a  weight  of  100  g. 
will  just  support  the 


I L 


EZt 


300  g.      This    point  FlG    139    The  gimple  leyer 

will  be  found  to  be 

45  cm.  from  the  fulcrum.    It  will  be  seen  at  once  that  the  product 

of  300  x  15  is  equal  to  the  product  of  100  x  45. 

Next  let  the  point  be  found  at  which  150  g.  just  balance  the  300  g. 
This  will  be  found  to  be  30  cm.  from  the  fulcrum.  Again,  the  products 
300  x  15  and  150  x  30  are  equal.  " 

No  matter  where  the  weights  are  placed,  or  what  weights 
are  used  on  either  side  of  the  fulcrum,  the  product  of  the 
effort  F  by  its  distance  I  from  the  fulcrum  (Fig.  140)  will  be 
found  to  be  equal  to  the  product  of  the  resistance  F'  by  its 
distance  lf  from  the  fulcrum.  Now  the  distances  I  and  I'  are 
called  the  lever  arms  of  the  forces  F  and  F',  and  the  product 


WORK  AND  THE  LEVER 


1ST 


of  a  force  by  its  lever  arm  is  called  the  moment  of  that  force. 
The  above  experiments  on  the  lever  may  then  be  generalized 
in  the  following  law :  The  ^  p  B 

moment  of  the  effort  is  equal 
to  the  moment  of  the  resist- 
ance. Algebraically  stated, 

it  is         pl  =  F'l'.          (4) 

, 
It  will  be  seen  that  the 


F' 


FIG.  140.  Illustrating  the  law  of  moments, 

namely  Fl  =  F'l' 
mechanical  advantage  of  the 

lever,  namely  F'/F,  is  equal  to  l/l'\  that  is,  to  the  lever  arm 

of  the  effort  divided  by  the  lever  arm  of  the  resistance. 

^/  179.  General  laws  of  the  lever.   If  parallel  forces  are  applied 

/at  several  points  on  a  lever,  as  in  Figs.  141  and  142,  it  will  be 

found,  in  the  particular  cases  illustrated,  that  for  equilibrium 

200  x  30  =100  x  20  +100  x  40 
and        300  x  20  +  50  x  40  =  100  x  15  +  200  x  32.5  ; 
that  is,  the  sum  of  all  the  moments  which  are  tending  to  turn  the 
beam  in  one  direction  is  equal  to  the  sum  of  all  the  moments  tend- 
ing to  turn  it  in  the  opposite  direction. 

If,  further,  we  support  the  levers  of  Figs.  141  and 
142  by  spring  balances  attached  at  P,  we,shall  find, 

A  P        B 

T 


iOO 


200 


CD 

300 


tr> 


FIG.  141  FIG.  142 

Condition  of  equilibrium  of  a  bar  acted  upon  by  several  forces 

after  allowing  for  the  weight  of  the  meter  stick,  that  the  two 
forces  indicated  by  the  balances  are  respectively  200  + 100  4- 
100  =  400  and  300  + 100  +  200  -  50  =  550  ;  that  is,  the  sum 
of  all  the  forces  acting  in  one  direction  on  the  lever  is  equal  to 
the  sum  of  all  the  forces  acting  in  the  opposite  direction. 


138  WOKK  AND  MECHANICAL  ENERGY 

These  two  laws  may  be  combined  as  follows :  If  we  think 
of  the  force  exerted  by  the  spring  balance  as  the  equilibrarit 
of  all  the  other  forces  acting  on  the  lever,  then  we  find  that  the 
resultant  of  any  number  of  parallel  forces  is  their  algebraic  sum, 
and  its  point  of  application  is  the  point  about  which  the  algebraic 
sum  of  the  moments  is  zero. 

180.  The  couple.    There   is   one   case,   however,   in  which 
parallel  forces  can  have  110  single  force  as  their  resultant, 
namely,  the  case  represented  in  Fig.  143.    Such  a  pair  of    j2 
equal  and  opposite  forces  acting  at  different  points  on  a 
lever  is  called  a  couple  and  can  be 

neutralized  only  by  another  couple 

tending  to  produce  rotation  in  the 

opposite  direction.    The  moment  of      ^    FlG" 143'  The  ( 

such  a  couple  is  evidently  F1  x  oa  +  F^  x  ob  =  Fl  x  ab ;  that  is, 

it  is  one  of  the  forces  times  the  total  distance  between  them. 

181.  Work  expended  upon  and  accomplished  by  the  lever. 
We  have  just  seen  that  when  the  lever  is  in  equilibrium  — 
that  is,  when  it  is  at  rest  or  is  moving  uniformly  —  the  relation 
between  the  effort  F  and  the  resistance  F1  is  expressed  in 
the    equation    of    moments, 

namely  Fl  =  F'l'.  Let  us 
now  suppose,  precisely  as  in 
the  case  of  the  pulleys,  that 
the  force  F  raises  the  weight 
F'  through  a  small  distance  FlG-  144-  Showing  that  the  equation 
f  rp  T  -i  ,1  •  ,1  of  moments,  Fl  =  F'l',  is  equivalent  to 

s.    To  accomplish  this,  the  '  Fs  _  p/s, 

point  A  to  which  F  is  at- 
tached must  move  through  a  distance  s  (Fig.  144).  From  the 
similarity  of  the  triangles  APn  and  BPm  it  will  be  seen  that 
l/V  is  equal  to  s/s'.  Hence  equation  (4),  which  represents  the 
law  of  the  lever,  and  which'  may  be  written  F/F'  =  I '  //,  may 
also  be  written  in  the  form 

F?F'  =  *'/*,  or  Fs  =  F's'. 


WORK  AND  THE 'LEVER  (  139/ 

Now  Fs  represents  the  work  done  by  the  effort  F,  and  F's' 
the  work  done  against  the  resistance  F'.  Hence  the  law  of 
moments,  which  has  just  been  found  by  experiment  to  be  the 
law  of  the  lever,  is  equivalent  to  the  statement  that  whenever 
work  is  accomplished  by  the  use  of  the  lever,  the  work  expended 
upon  the  lever  by  the  effort  F  is  equal  to  the  work  accomplished 
by  the  lever  against  the  resistance  F'. 

182.  The  three  classes  of  levers.  It  is  customary  to  divide 
levers  into  three  classes,  as  follows : 

1.  In  levers  of  the  first  class  the  fulcrum  P  is  between  the 
acting  force  F  and  the  resisting  force  F'  (Fig.  145).  The 


fp 

i 

____^v  

-x 

P 

I 
__—  ^—  _ 

F\ 

L^ 
& 

(1) 

1 
F 

^ 

F'     (1) 

(2) 

(2) 

F'«P 


P    Fr 


FIG.  145.   Levers  of 
first  class 


FIG.  146.    Levers  of 
second  class 


FIG.  147.   Levers  of 
third  class 


mechanical  advantage  of  levers  of  this  class  is  greater  or  less 
than  unity  according  as  the  lever  arm  I  of  the  effort  is  greater 
or  less  than  the  lever  arm  I'  of  the  resistance. 

2.  In  levers  of  the  second  class  the  resistance  F'  is  between 
the  effort  F  and  the  fulcrum  P  (Fig.  146).    Here  the  lever 
arm  of  the  effort,  that  is,  the  distance  from  F  to  P,  is  neces- 
sarily greater  than  the  lever  arm  of  the  resistance,  that  is,  the 
distance  from  F'  to  P.    Hence  the  mechanical  advantage  is 
always  greater  than  1. 

3.  In  levers  of  the  third  class  the  acting  force  is  between  the 
resisting  force  and  the  fulcrum  (Fig.  147).    The  mechanical 
advantage  is  then  obviously  less  than  1,  that  is,  in  this  type 
of  lever  force  is  always  sacrificed*  for  the  sake  of  gaining  speed. 


140 


WOKK  AND  MECHANICAL  ENERGY 


QUESTIONS  AND  PROBLEMS 

1.  Two  boys  are  carrying  a  bag  of  walnuts  at  the  middle  of  a 
long  stick.    Will  it  make  any  difference  whether  they  walk  close  to 
the  bag  or  farther   away,  so  long  as  each  is  at  the   same   distance? 

2.  When  a  load  is  carried  on   a  stick  over  the  shoulder,  why  does 
the  pressure  on  the  shoulder  become 

greater  as  the  load  is  moved  farther 
out  on  the  stick  ? 

3.  Explain  the  use  of  the  rider  in 
weighing  (see  Fig.  23). 

4.  In  which  of  the  three  classes  of 
levers  does  the  wheelbarrow  belong? 
grocer's  scales?  pliers?  sugar  tongs? 
a  claw  hammer?  a  pump  handle? 

5.  Explain  the  principle  of  weigh- 

.T..  FIG.  148.    Steelyards 

ing  by  the  steelyards  (Fig.  148).  What 

must  be  the  weight  of  the  bob  P  if,  at  a  distance  of  40  cm.  from 
the  fulcrum  0,  it  balances  a  weight  of  10  kg.  placed  at  a  distance  of 
2  cm.  from  0  ? 

6.  If  you  knew  your  own  weight,  how  could  you  determine  the 
weight  of  a  companion  if  you  had  only  a  teeter  board  and  a  foot  rule  ? 

7.  How  would   you    arrange  a  crowbar  to    use    it  as   a   lever  of 
the  first  class  in  overturning  a  heavy  object  ?  as  a  lever  of  the  second 
class? 

8.  If  3  horses  are  to  pull  equally  on  a  load,  how  should  the  whipple- 
tree  be  designed? 

9.  Two  boys  carry  a  load  of  60  Ib.  on  a  pole  between  them.    If  the 
load  is  4  ft.  from  one  boy  and  6  ft.  from  the  other,  how  many  pounds 
does  each  boy  carry?    (Consider  the 

force  exerted  by  one  of  the  boys 
as  the  effort,  the  load  as  the  resist- 
ance, and  the  second  boy  as  the 
fulcrum.) 

10.  Why  is  it  that  a  couple  can- 
not be  balanced  by  a  single  force  ? 

11.  Why  do  tinners'  shears  have 
long  handles   and   short  blades   and 
tailors'  shears  just  the  opposite? 

12.  If  the  ball  of  the  float  valve  (Fig.  149)  has  a  diameter  of  10  cm., 
and  if  the  distance  from  the  center  of  the  ball  to  the  pivot  S  is  20 
times  the  distance  from  S  to  the  pin  P,  with  what  force  is  the  valve 
R  held  shut  when  the  ball  is  half  immersed? 


FIG.  149.    The  automatic 
float  valve 


THE  PRINCIPLE  OF  WORK 


141 


13.  In  the  Yale  lock  (Fig.  150)  the  cylinder  G  rotates  inside  the 
fixed  cylinder  F  and  works  the  bolt  through  the  arm  H.  The  right  key 
raises  the  pins  «',  />',  c',  d',  e',  until  their  tops  are  just  even  with  the  top 
of  G.  What  mechanical  principles  do  you  find  involved  in  this  device  ? 

(1)  (2) 


FIG.  150.    Yale  lock 
(1)  The  right  key  ;    (2)  the  wrong  key 

t 

THE  PRINCIPLE  OF  WORK 

183.  Statement  of  the  principle  of  work.  The  study  of 
pulleys  led  us  to  the  conclusion  that  in  all  cases  where  such 
machines  are  used,  the  work  done  by  the  effort  is  equal  to  the 
work  done  against  the  resistance,  provided  always  that  friction 
may  be  neglected,  and  that  the  motions  are  uniform  so  that 
none  of  the  force  exerted  is  used  in  overcoming  inertia.  The 
study  of  levers  led  to  precisely  the  same  result.  In  Chapter  II 
the  study  of  the  hydraulic  press  showed  that  the  same  law 
applied  in  this  case  also,  for  it  was  shown  that  the  force  on 
the  small  piston  times  the  distance  through  which  it  moved 
was  equal  to  the  force  on  the  large  piston  times  the  distance 
through  which  it  moved.  Similar  experiments  upon  all  sorts 
of  machines  have  shown  that  in  all  cases  where  friction  may 
be  neglected  the  following  is  an  absolutely  general  law:  In 
all  mechanical  devices  of  whatever  sort  the  work  ^expended  'upon 
the  machine  is  equal  to  the  work  accomplished  by  it. 

This  important  generalization,  called  "  the  principle  of 
work,"  was  first  stated  by  Newton  in  1687.  It  has  proved  to 
be  one  of  the  most  fruitful  principles  ever  put  forward  in  the 
history  of  physics.  By  its  application  it  is  easy  to  deduce  the 
relation  between  the  force  applied  and  the  force  overcome  in 


142 


WORK'  AND  MECHANICAL  ENEEGY 


any  sort  of  machine,  provided  only  that  friction  is  negligible, 
and  that  the  motions  take  place  slowly.  It  is  only  necessary 
to  produce,  or  imagine,  a  displacement  at  one  end  of  the 
machine,  and  then  to  measure  or  calculate  the  corresponding 
displacement  at  the  other  end.  The  ratio  of 
the  second  displacement  to  the  first  is  the 
ratio  of  the  force  acting  to  the  force  overcome. 
184.  The  wheel  and  axle.  Let  us  apply  the 
work  principle  to  discover  the  law  of  the  wheel 
and  axle  (Fig.  151).  When  the  large  wheel 
has  made  one  revolution,  the  point  A  moves 
down  a  distance  equal  to  the  circumference  of 
this  wheel.  During  this  time  the  weight  F1  is 
lifted  a  distance  equal  to  the  circumference  of 
the  axle.  Hence  the  equation  Fs  =  F's'  be- 
comes F  x  2  TrR  =  F'  x  2  TIT,  where  R  and  r  are  the  radii  of 
the  wheel  and  axle  respectively.  This  equation  may  be 
written  in  the  form  pl/F  =  R/r  . 


FIG.  151.    The 
wheel  and  axle 


that  is,  the  weight  lifted  on  the  axle  is  as  many  times  the  force 
applied  to  the  wheel  as  the  radius  of  the  wheel  is  times  the  radius 
of  the  axle.  Otherwise  stated,  the  me- 
chanical advantage  of  the  wheel  and 
axle  is  equal  to  the  radius  of  the  wheel 
divided  by  the  radius  of  the  axle. 

The  capstan  (Fig.  152)  is  a  spe- 
cial case  of  the  wheel  and  axle,  the 
length  of  the  lever  arm  taking  the 
place  of  the  radius  of  the  wheel, 
and  the  radius  of  the  barrel  corre- 
sponding to  the  radius  of  the  axle.  FlG'  152'  The  capstan 

185.  The  work  principle  applied  to  the  inclined  plane.  The 
work  done  against  gravity  in  lifting  a  weight  F'  (Fig.  153) 
from  the  bottom  to  the  top  of  a  plane  is  evidently  equal 


THE  PRINCIPLE  OF  WORK 


143 


to  F'  times  the  height  h  of  the  plane.  But  the  work  done  by 
the  acting  force  F,  while  the  carriage  of  weight  F'  is  being 
pulled  from  the  bottom  to  the 
top  of  the  plane,  is  equal  to  F 
times  the  length  I  of  the  plane. 
Hence  the  principle  of  work  gives 

Fl  =  F'h,  or  F'/F  =  l/h  ;    (6) 

FIG.  153.    The  inclined  plane 
that  is,  the  mechanical  advantage  of 

the  inclined  plane,  or  the  ratio  of  the  weight  lifted  to  the  force 
acting  parallel  to  the  plane,  is  the  ratio  of  the  length  of  the  plane 
to  the  height  of  the  plane.  This  is  precisely  the  conclusion 
at  which  we  arrived  in  another  way  in  Chap- 
ter V,  p.  80. 

186.  The  screw.  The  screw  (Fig.  154)  is  a 
combination  of  the  inclined  plane  and  the 
lever.  Its  law  is  easily  obtained  from  the  prin- 
ciple of  work.  When  the  force  which  acts  on 
the  end  of  the  lever  has  moved  this  point 
through  one  complete  revolution,  the  weight 
F',  which  rests  on  top  of  the  screw,  has  evi- 
dently been  lifted  through  a  vertical  distance 
equal  to  the  distance  between  two  adjoining  threads.  This 
distance  d  is  called  the  pitch  of1  the  screw.  Hence,  if  we 
represent  by  I  the  length  of  the  lever, 
the  work  principle  gives 


FIG.  154.   The 
jackscrew 


27rl  =  F'd ; 


(7) 


that   is,    the    mechanical    advantage    of 

the  screw,  or  ratio  of  the  weight  lifted 

to    the  force    applied,    is    equal    to    the 

ratio  of  the  circumference  of  the   circle 

moved  over  by  the  end  of  the  lever,  to  the  distance  between  the 

threads  of  the  screw.    In  actual  practice  the  friction  in  such 


FIG.  155.  The  letter  press 


144 


WORK  AND  MECHANICAL  ENERGY 


an  arrangement  is  always  very  great,  so  that  the  mechanical 
advantage  is  considerably  less  than  its  full  theoretical  value. 
The  common  jackscrew  just  described  (and 
used  chiefly  for  raising  buildings),  the  letter 
press  (Fig.  155),  and  the  vise  (Fig.  156)  are 
all  familiar  forms  of  the  screw. 

187.  A  train  of  gear  wheels.    A  form  of  machine   _ 

.  *  FIG.  156.   The  vise 

capable  of  very  high  mechanical  advantage  is  the 

train  of  gear  wheels  shown  in  Fig.  157.   Let  the  student  show  from  the 
principle  of  work,  namely  Fs  =  F's',  that  the  mechanical  advantage, 

F' 

that  is,  — ,  of  such  a  device  is 
F 

,         no.  cogs  in  d      no.  cogs  in  /' 

circum.  of  a  x  — • —       x  — 

no.  cogs  in  c      no.  cogs  111  b 

circum.  of  e 

188.  The  worm  wheel.   Another  device  of  high  mechanical  advantage 
is  the  worm  wheel  (Fig.  158).    Show  that  if  /  is  the  length  of  the  crank 
arm  C,  n  the  number 

of  teeth  in  the  cog- 
wheel W,  and  r  the 
radius  of  the  axle,  the 
mechanical  advantage 
is  given  by 


FIG.  157.   Train  of  gear 
wheels 


FIG.  158.   The  worm 
gear 


(9) 


This  device  is  used 
most  frequently  when 
the  primary  object  is 

to  decrease  speed  rather  than  to  multiply  force.    It  will  be  seen  that  the 
crank  handle  must  make  n  turns  while  the  cogwheel  is  making  one. 

189.  The  differential  pulley.  In  the  differential  pulley  (Fig.  159)  an 
endless  chain  passes  first  over  the  fixed  pulley  A,  then  down  and  around 
the  movable  pulley  C,  then  up  again  over  the  fixed  pulley  B,  which  is 
rigidly  attached  to  A,  but  differs  slightly  from  it  in  diameter.  On  the 
circumference  of  all  the  pulleys  are  projections  which  fit  between  the 
links,  and  thus  keep  the  chains  from  slipping.  When  the  chain  is 
pulled  down  at  F,  as  in  Fig.  159  (2),  until  the  upper  rigid  system  of 


THE  PRINCIPLE  OF  WORK 


145 


pulleys  has  made  one  complete  revolution,  the  chain  between  the  upper 
and  lower  pulleys  has  been  shortened  by  the  difference  between  the 
circumferences  of  the  pulleys  A 
and  B,  for  the  chain  has  been 
pulled  up  a  distance  equal  to 
the  circumference  of  the  larger 
pulley  and  let  down  a  distance 
equal  to  the  circumference  of  the 
smaller  pulley.  Hence  the  load 
F"  has  been  lifted  by  half  the 
difference  between  the  circumfer- 
ences of  A  and  B.  The  mechan- 
ical advantage  is  therefore  equal 
to  the  circumference  of  A  divided 
by  one  half  the  difference  between 
the  circumferences  of  A  and  B. 

FIG.  159.    The  differential  pulley 

QUESTIONS  AND  PROBLEMS 

1.  A  300-lb.  barrel  was  rolled  up  a  plank  12  ft.  long  into  a  doorway 
3  ft.  high.    What  force  was  applied  parallel  to  the  plank  ? 

2.  A  1500-lb.  safe  must  be  raised  5  ft.    The  force  which  can  be 
applied  is  250  Ib.    What  is  the  shortest  inclined  plane  which  can  be 
used  for  the  purpose  ? 

3.  In  the  differen- 
tial   wheel    and    axle 
(Fig.  160)  the  rope  is 
wound  in  opposite  di- 
rections on  two  axles 
of  different  diameter. 
For  a  complete  revo- 
lution of  the  axle  the 
weight  is  lifted  by  a 
distance  equal  to  one 
half  the  difference  be- 
tween the  circumfer- 
ences of  the  two  axles.   If  the  crank  has  a  radius  of  2  ft.,  the  larger  axle 
a  diameter  of  6  in.,  and  the  smaller  one  a  diameter  of  4  in.,  find  the 
mechanical  advantage  of  the  arrangement. 

4.  If,  in  the  compound  lever  of  Fig.  161,  A  C=  6  ft.,  EC  =  'l  ft., 
DE  =  4  ft.,  EG  =  8  in.,  HJ=5  ft.,  and  IJ=2  ft.,  what  force  applied 
at  F  will  support  a  weight  of  2000  Ib.  at  F'  ? 

t 


FIG.  160.    Differential 
windlass 


FIG.  161.    The  com- 
pound lever 


146 


WOEK  AND  MECHANICAL  ENERGY 


5.  The  hay  scales  shown  in  Fig.  162  consist  of  a  compound  lever  with 
fulcrums  at  F,  F',  F",  F'".  If  Fo  and  F'o'  are  lengths  of  6  in.,  FE  and 
F'E  5  ft.,  F"n  1  ft.,  ~P"m  6  ft.,  rF'"  2  in.,  and  F'"S  20  in.,  how  many 
pounds  at  W  will  be  required  to  balance  a  weight  of  a  ton  on  the  platform  ? 


FIG.  162.    Hay  scales 


FIG.  163.    Windlass  with 
gears 


6.  If  the  capstan  of  a  ship  is  12  in.  in  diameter  and  the  levers  are 
6  ft.  long,  what  force  must  be  exerted  by  each  of  4  men  in  order  to  raise 
an  anchor  weighing  2000  lb.? 

7.  In  the  windlass  of  Fig.  163  the  crank  handle  has  a  length  of 
2  ft.,  and  the  barrel  a  diameter  of  8  in.   There  are  20  cogs  in  the  small 
cogwheel  and  60  in  the 

large  one.  What  is  the 
mechanical  advantage 
of  the  arrangement? 

8.  A  force  of  80  kg. 
on  a  wheel  whose  diam- 
eter ,is  3  m.  balances  a 
weight  of  150  kg.  on  the 
axle.  Find  the  diameter 
of  the  axle. 

9.  Eight  jackscrews 
each    of    which    has    a 
pitch  of  i-  in.  and  a  lever 
arm  of  18  in.  are  being 
worked    simultaneously 
to  raise  a  building  weigh- 
ing 100,000  lb.   What  force  would  have  to  be  exerted  at  the  end  of  each 
lever  if  there  were  no  friction?    What  if  75%  were  wasted  in  friction? 

10.  If  a  worm  wheel  (Fig.  158)  has  30  teeth,  and  the  crank  is 
25  cm.  long,  while  the  radius  of  the  axle  is  3  cm.,  what  is  the  mechan- 
ical advantage  of  the  arrangement? 


POWER  AND  ENERGY 


14T 


11.  If  in  the  crane  of  Fig.  164  the  crank  arm  has  a  length  of  1/2  m., 
and  the  gear  wheels  A,  B,  C,  and  D  have  respectively  12,  48,  12,  and  60 
cogs,  while  the  axle  over  which 

the  chain  runs  has  a  radius  of 
10  cm.,  what  is  the  mechanical 
advantage  of  the  crane? 

12.  With  the  aid  of  Fig.  165 
describe  the    process   of    winding 
and  setting  a  watch.    The  rocker 
R  is  pivoted  at  S.    C  carries  the 
mainspring    and    E    the    hands. 
S.P.  is  a  light  spring  which  nor- 
mally keeps  the  wheel  A  in  mesh 
with  C.  Pressing  down  oji  P,  how- 
ever, releases  A  from  C  and  en- 
gages B  with  D.  What  mechanical 

principles  do  you  find  involved  ?    FIG.  165.  Winding  and  setting  mech- 


What  happens  when  M  is  turned 
backward  ? 


anism  of  a  stem-winding  watch 


POWER  AND  ENERGY 

190.  Definition  of   power.    When   a  given   load  has  been 
raised  a  given   distance  a  given   amount  of  work  has  been 
done,  whether  the  time  consumed  in  doing  it  is  small  or  great. 
Time  is  therefore  not  a  factor  which  enters  into  the  deter- 
mination of  work;  but  it  is  often  as  important  to  know  the 
rate  at  which  work  is  done  as  to  know  the  amount  of  work 
accomplished.    The  rate  of  doing  work  is  catted  power,  or  activity. 
Thus,  if  P  represent  power,  W  the  work  done,  and  t  the  time 
required  to  do  it,  w 

P-f  (10) 

191.  Horsepower.    James  Watt  (1736-1819),  the  inventor 
of  the  steam  engine,  considered  that  an  average  horse  could  do 
33,000  foot  pounds  of  work  per  minute,  or  550  foot  pounds  per 
second.     The   metric    equivalent   is    76.05   kilogram  meters 
per  second.  This  number  is  probably  considerably  too  high,  but 
it  has  been  taken  ever  since,  in  English-speaking  countries, 


148 


WOEK  AND  MECHANICAL  ENERGY 


as  the  unit  of  power,  and  named  the  horse  power  (H.P.). 
The  power  of  steam  engines  has  usually  been  rated  in  horse 
power.  The  horse  power  of  an  ordinary  railroad  locomotive  is 
from  500  to  1000.  Stationary  engines  and  steamboat  engines 
of  the  largest  size  often  run  from  5000  to  20,000  H.P.  The 
power  of  an  average  horse  is  about  3/4  H.P.,  and  that  of  an 
ordinary  man  abqut  1/7  H.P. 

192.  The  kilowatt.     In  the  metric  system  the  erg  has  been 
taken  as  the  absolute  unit  of  work.    The  corresponding  unit  of 
power  is  an  erg  per  second.    This  is,  however,  so  small  that  it 
is  customary  to  take  as  the  practical  unit  10,000,000  ergs  per 
second;  that  is,  one  joule  per  second  (see  §  173,  p.  132).   This 
unit  is  called  the  watt,  in  honor  of  James  Watt.    The  power 
of  dynamos  and  electric  motors  is  almost  always  expressed  in 
kilowatts,  a  kilowatt  representing  1000  watts,  and  in  modern 
practice  even  steam  engines  are  being  increasingly  rated  in 
kilowatts  rather  than  in  horse  power.   A  horse  power  is  equiva- 
lent to  746  watts ;  it  may  therefore  in  general  be  considered 
to  be  3/4  of  a  kilowatt. 

193.  Definition  of  energy.    The  energy  of  a 
body  is  defined  as  its  capacity  for  doing  work. 
In  general,  inanimate  bodies  possess  energy 
only  because  of  work  which  has  been  done 
upon  them  at  some  previous  time.     Thus, 
suppose  a  kilogram  weight  is  lifted  from  the 
first  position  in  Fig.  166  through  a  height  of 
1  m.,  and  placed  upon  the  hook  H  at  the  end 
of  a  cord  which  passes  over  a  frictionless 

pulley  p  and  is  attached  at  the  other  end  to    FlG- 166-  lllustra- 

a  second  kilogram  weight  B.    The  operation 

of  lifting  A  from  position  1  to  position  2  has 

required  an  expenditure  upon  it  of  1  kg.  m.  (100,000  g.  cm., 

or  98,000,000  ergs)  of  work.    But  in  position  2,  A  is  itself 

possessed  of  a  certain  capacity  for  doing  work  which  it  did 


tion  of    potential 
energy 


JAMES  PRESCOTT  JOULE 

(1818-1889) 

English,  physicist,  born  at  Man- 
chester ;  most  prominent  figure  in 
the  establishment  of  the  doctrine 
of  the  conservation  of  energy ; 
studied  chemistry  as  a  boy  under 
John  Dalton,  and  became  so  inter- 
ested that  his  father,  a  prosperous 
Manchester  brewer,  fitted  out  a 
laboratory  for  him  at  home  ;  con- 
ducted mostof  his  researches  either 
in  a  basement  of  his  own  house  or 
in  a  yard  adjoining  his  brewery ; 
discovered  the  law  of  heating  a 
conductor  by  an  electric  current ; 
carried  out,  in  connection  with 
Lord  Kelvin,  epoch-making  re- 
searches upon  the  thermal  prop- 
erties of  gases;  did  important  work 
in  magnetism;  first  proved  experi- 
mentally the  identity  of  various 
forms  of  energy 


JAMES  WATT  (1736-1819) 

The  Scotch  instrument  maker  at 
the  University  of  Glasgow,  Avho 
may  properly  be  considered  the 
inventor  of  the  steam  engine  ;  for, 
although  a  crude  and  inefficient 
type  of  steam  engine  was  known 
before  his  time,  he  left  it  in  essen- 
tially its  present  form.  The  mod- 
ern industrial  era  may  be  said  to 
begin  with  Watt 


POWER  AND  ENERGY  149 

not  have  before.  For  if  it  is  now  started  downward  by  the 
application  of  the  slightest  conceivable  force,  it  will,  of  its 
own  accord,  return  to  position  1,  and  will  in  so  doing  raise 
the  kilogram  weight  B  through  a  height  of  1  m.  In  other 
words,  it  will  do  upon  B  exactly  the  same  amount  of  work 
which  was  originally  done  upon  it. 

194.  Potential  and  kinetic  energy.    A  body  may  have  a 
capacity  for  doing  work  not  only  because  it  has  been  given  an 
elevated  position,  but  also  because  it  has  in  some  way  acquired 
velocity ;  for  example,  a  heavy  flywheel  will  keep  machinery 
running  for  some  time  after  the  power  has  been  shut  off ;  a  bul- 
let shot  upward  will  lift  itself  a  great  distance  against  gravity 
because  of  the  velocity  which  has  been  imparted  to  it.    Simi- 
larly, any  body  which  is  in  motion  is  able  to  rise  against  grav- 
ity, or  to  set  other  bodies  in  motion  by  colliding  with  them, 
or  to  overcome  resistances  of  any  conceivable  sort.    Hence,  in 
order  to  distinguish  between  the  energy  which  a  body  may 
have  because  of  an  advantageous  position,  and  the  energy  which 
it  may  have  because  it  is  in  motion,  the  two  terms  "  potential " 
and  "  kinetic  "  energy  are  used.   Potential  energy  includes  the 
energy  of  lifted  weights,  of  coiled  or  stretched  springs,  of  bent 
bows,  etc. ;  in  a  word,  potential  energy  is  energy  of  position,  while 
kinetic  energy  is  energy  of  motion. 

195.  Transformations  oi  potential  and  kinetic  energy.    The 
swinging   of    a   pendulum   and   the   oscillation   of   a  weight 
attached  to  a  spring  illustrate  well  the  way  in  which  energy 
which  has  once  been  put  into  a  body  may  be  transformed  back 
and  forth  between  the  potential  and  kineti^c  varieties.    When 
the  pendulum  bob  is  at  rest  at  the  bottom  of  its  arc  it  pos- 
sesses no  energy  of  either  type,  since,  on  the  one  hand,  it  is 
as  low  as  it  can  be,  and,  on  the  other,  it  has  no  velocity.  When 
we  pull  it  up  the  arc  to  the  position  A  (Fig.  167),  we  do  an 
amount  of  work  upon  it  which  is  equal  in  gram  centimeters 
to  its  weight  in  grams  times  the  distance  AD  in  centimeters ; 


150  WOEK  AHD  MECHANICAL  EHEKGY 

that  is,  we  store  up  in  it  this  amount  of  potential  energy.  As 
now  the  bob  falls  to  C  this  potential  energy  is  completely 
transformed  into  kinetic.  That  this  kinetic  energy  at  C  is 
exactly  equal  to  the  potential  energy 
at  A  is  proved  by  the  fact  that  if  fric- 
tion is  completely  eliminated,  the  bob 
rises  to  a  point  B  such  that  BE  is 
equal  to  AD.  We  see,  therefore,  that 
at  the  ends  of  its  swing  the  energy  of 
the  pendulum  is  all  potential,  while 
in  the  middle  of  the  swing  its  energy 
is  all  kinetic.  In  intermediate  posi- 
tions the  energy  is  part  potential 
and  part  kinetic,  but  the  sum  of  the  FIG.  167.  Transformation 

two  is  equal  to  the  original  potential     of  P°tential  and  kinetic 

energy 
energy. 

196.  General  statement  of  the  law  of  frictionless  machines. 
In  our  development  of  the  law  of  machines,  which  led  us  to 
the  conclusion  that  the  work  of  the  acting  force  is  always 
equal  to  the  work  of  the  resisting  force,  we  were  careful  to 
make  two  important  assumptions : '  first,  that  friction  was 
negligible ;  and  second,  that  the  motions  were  all  either  uni- 
form or  so  slow  that  no  appreciable  velocities  were  imparted. 
In  other  words,  we  assumed  that  the  work  of  the  acting  force 
was  expended  simply  in  lifting  weights  or  compressing  springs; 
that  is,  in  storing  up  potential  energy.  If  now  we  drop  the 
second  assumption,  a  very  simple  experiment  will  show  that 
our  conclusion  n\ust  be  somewhat  modified.  Suppose,  for 
instance,  that  instead  of  lifting  a  500-g.  weight  slowly  by 
means  of  a  balance,  we  jerk  it  up  suddenly.  We  shall  now 
find  that  the  initial  pull  indicated  by  the  balance,  instead  qf 
being  500  g.,  will  be  considerably  more  —  perhaps  as  much 
as  several  thousand  grams  if  the  pull  is  sufficiently  sudden. 
This  is  obviously  because  the  acting  force  is  now  overcoming 


POWER,  AND  ENERGY  151 

not  merely  the  500  g.  which  represents  the  resistance  of  grav- 
ity, but  also  the  inertia  of  the  body,  since  velocity  is  being 
imparted  to  it.  Now  work  done  in  imparting  velocity  to  a 
body,  that  is,  in  overcoming  its  inertia,  always  appears  as 
kinetic  energy,  while  work  done  in  overcoming  gravity  appears 
as  the  potential  energy  of  a  lifted  weight.  Hence,  whether  the 
motions  produced  by  machines  are  slow  or  fast,  if  friction  is 
negligible,  the  law  for  all  devices  for  transforming  work  may 
be  stated  thus :  The  work  of  the  acting  force  is  equal  to  the  sum 
of  the  potential  and  kinetic  energies  stored  up  in  the  mass  acted 
upon.  In  machines  which  work  against  gravity  the  body  usually 
starts  from  rest  and  is  left  at  rest,  so  that  the  kinetic  energy 
resulting  from  the  whole  operation  is  zero.  Hence  in  such  cases 
the  work  done  is  the  weight  lifted  times  the  height  through 
which  it  is  lifted,  whether  the  motion  is  slow  or  fast.  The 
kinetic  energy  imparted  to. the  body  in  starting  is  all  given 
up  by  it  in  stopping. 

197.  The  measure  of  potential  and  kinetic  energy.  The 
measure  of  the  potential  energy  of  any  lifted  body,  such  as  a 
lifted  pile  driver,  is  equal  to  the  work  which  has  been  spent 
in  lifting  the  body.  Thus  if  A  is  the  height  in  centimeters 
and  M  the  weight  in  grams,  then  the  potential  energy  P.E. 
of  the  lifted  mass  is 

P.E.  =  Mh  gram  centimeters.  (11) 

Since  the  force  of  the  earth's  attraction  for  M  grams  is  My  dynes, 
if  we  wish  to  express  the  potential  energy  in  ergs  instead  of  in  gram 
centimeters,  we  have  RR  =  Mgh  ergg>  (12) 

Since  this  energy  is  all  transformed  into  kinetic  energy  when  the 
mass  falls  the  distance  h,  the  product  Mgh  also  represents  the  number 
of  ergs  of  kinetic  energy  which  the  moving  weight  has  when  it  strikes 
the  pile. 

If  we  wish  to  express  this  kinetic  energy  in  terms  of  the  velocity  with 
which  the  weight  strikes  the  pile,  instead  of  the  height  from  which  it 
has  fallen,  we  have  only  to  substitute  for  h  its  value  in  terms  of  g  and 


152     WOEK  AND  MECHANICAL  ENEEGY 

the  velocity  acquired  [see  equation  (3),  p.  93],  namely  h  =  v^/2  g.   This 
gives  the  kinetic  energy  K.E.  in  the  form 

K.E.  =  J  Mv2  ergs.  (13) 

Since  it  makes  no  difference  how  a  body  has  acquired  its  velocity, 
this  represents  the  general  formula  for  the  kinetic  energy  in  ergs  of  any 
moving  body,  in  terms  of  its  mass  and  its  velocity. 

Thus  the  kinetic  energy  of  a  100-g.  bullet  moving  with  a  velocity  of 
10,000  cm.  per  second  is 

K.E.  =  £  x  100  x  (10,000)2  =  5,000,000,000  ergs. 

Since  1  g.  cm.  is  equivalent  to  980  ergs,  the  energy  of  this  bullet  is 
5'000908°o0'000  =  5,102,000  g.  cm.,  or  51.02  kg.  m. 

We  know,  therefore,  that  the  powder  pushing  on  the  bullet  as  it 
moved  through  the  rifle  barrel  did  51.02  kg.  m.  of  work  upon  the  bullet 
in  giving  it  the  velocity  of  100  m.  per  second. 

In  general  terms,  if  M  is  in  grams  and  v  in  centimeters  per  second, 


K.E.  =  —  -  g.  cm.  ;   if  M  is  in  pounds  and  v  in  feet  per  second, 

K.E.  =       Mv*      ft.  Ib. 
2  x  32.16 

QUESTIONS  AND  PROBLEMS 

1.  A  150-lb.  man  runs  up  a  flight  of  stairs  60  ft.  high  in  10  sec.   What 
is  his  horse  power  while  doing  it?    How  do  you  account  for  the  result? 

2.  If  a  rifle  bullet  can  just  pass  through  a  plank,  how  many  planks 
will  it  pass  through  if  its  speed  is  doubled? 

3.  What  must  be  the  horse  power  of  an  engine  which  is  to  pump 
10,000  1.  of  water  per  second  from  a  mine  150  m.  deep  ? 

4.  What  must  be  the  power  in  kilowatts  of  the  engines  supplying 
the  city  water  in  Problem  4,  p.  132  ?    Express  the  power  also  in  horse 
power.    (Assume  a  24-hour  day.) 

5.  A  water  motor  discharges  100  1.  of  water  per  minute  when  fed  from 
a  reservoir  in  which  the  water  surface  stands  50  in.  above  the  level  of 
the  motor.    If  all  of  the  potential  energy  of  the  water  were  transformed 
into  work  in  the  motor,  what  would  be  the  horse  power  of  the  motor  ? 
(The  potential  energy  of  the  water  is  the  amount  of  work  which  would 
be  required  to  carry  it  back  to  the  top  of  the  reservoir.) 


CHAPTER  IX 

WORK  AND  HEAT  ENERGY 
FRICTION 

198.  Friction  always  results  in  wasted  work.    All  of  the 

experiments  mentioned  in  the  last  chapter  were  so  arranged 
that  friction  could  be  neglected  or  eliminated.  So  long  as  this 
condition  was  fulfilled  it  was  found  that  the  result  of  uni- 
versal experience  could  be  stated  thus :  The  work  done  by 
the  acting  force  is  equal  to  the  sum  of  the  kinetic  and  potential 
energies  stored  up. 

But  wherever  friction  is  present  this* law  is  found  to  be  in- 
exact, for  the  work  of  the  acting  force  is  then  always  somewhat 
greater  than  the  sum  of  the  kinetic  and  potential  energies 
stored  up.  If,  for  example,  a  block  is  pulled  over  the  horizon- 
tal surface  of  a  table,  at  the  end  of  the  motion  no  velocity  has 
been  imparted  to  the  block,  and  hence  no  kinetic  energy  has 
been  stored  up.  Further,  the  block  has  not!  been  lifted  nor  put 
into  a  condition  of  elastic  strain,  and  hence  no  potential  energy 
has  been  communicated  to  it.  We  cannot  in  any  way  obtain 
from  the  block  more  work  after  the  motion  than  we  could  have 
obtained  before  it  was  moved.  It  is  clear,  therefore,  that  all 
of  the  work  which  was  done  in  moving  the  block  against  the 
friction  of  the  table  was  wasted  work.  Experience  shows  that, 
in  general,  where  work  is  done  against  friction  it  can  never 
be  regained.  Before  considering  what  becomes  of  this  wasted 
work  we  shall  consider  some  of  the  factors  on  which  friction 
depends,  and  some  of  the  laws  which  are  found  by  experiment 
to  hold  in  cases  in  which  friction  occurs. 

153 


154 


WOEK  AND  HEAT  ENERGY 


199.  Coefficient  of  friction.    It  is  found  that  if  F  represents 
the  force  parallel  to  a  plane  which  is  necessary  to  maintain 
uniform  motion  in  a  body  which  is  pressed  against  the  plane 

F 

with  a  force  F1,  then  the  ratio  —  depends  only  on  the  nature 

of  the  surfaces  in  contact,  and  not  at  all  on  the  area  or  on  the 

77* 

velocity  of  the  motion.    The  ratio  —  is  called  the  coefficient  of 

F 

friction  for  the  given  materials.  Thus,  if  F  is  300  g.  and  F' 
800  g.,  the  coefficient  of  friction  is  |^|  =  .375.  The  coefficient 
of  iron  on  iron  is  about  .2,  of  oak  on  oak  about  .4. 

200.  Rolling  friction.    The  chief  cause  of  sliding  friction  is  the  inter- 
locking of  minute  projections  (shown  greatly  magnified  at  a,  b,  c,  and  d 
in  Fig.  168).    When   a   round  solid 

rolls  over  a  smooth  surface  the  fric- 

tional  resistance  is  generally  much 

less  than  when  it  slides ;  for  example, 

the  coefficient  of  friction  of  cast-iron 

wheels  rolling  on  iron  rails  may  be 

as  low  as  .002 ;  that  is,  -^~  of  the  sliding  friction  of  iron  on  iron. 

This  means  that  a  pull  of  1  pound  will  keep  a  500-pound  car  in  motion. 

Sliding  friction  is  not,  however,  entirely  dispensed  with  in  ordinary 

(1)  (2) 


FIG.  168.    Illustrating  friction  of 
rubbing  surfaces 


FIG.  169.   Friction  in  bearings 
(1)  Common  bearing ;  (2)  ball  bearing 

wheels,  for  although  the  rim  of  the  wheel  rolls  on  the  track,  the  axle 
slides  continuously  at  some  point  c  [Fig.  169,  (1)]  upon  the  surface  of 
the  journal. 

The  great  advantage  of  the  ball  bearing  [Fig.  169,  (2)]  is  that  the 
sliding  friction  in  the  hub  is  almost  completely  replaced  by  rolling 
friction.  The  manner  in  which  ball  bearings  are  used  in  a  bicycle 


FRIOTIOK 


155 


pedal  is  illustrated  in  Fig.  170.  The  "free  wheel"  ratchet  is  shown  in 
Fig.  171.  The  "pawls"  a,  b,  enable  the  pedals  and  chain  wheel  W  to 
stop  while  the  rear  axle  continues  to  revolve. 

201.   Fluid  friction.    When  a  solid  moves  through  a  fluid,  as  when  a 
bullet  moves  through  the  air  or  a  ship  through  the  water,  the  resistance 
encountered  is  not  at  all  independent  of  veloc- 
ity, as  in  the  case  of  solid  friction,  but  increases 


FIG.  170.   The  bicycle  pedal 


FIG.  171.  Free  wheel  ratchet 


for  slow  speeds  nearly  as  the  square  of  the  velocity,  and  for  high  speeds 
at  a  rate  considerably  greater.  This  explains  why  it  is  so  expensive  to 
run  a  fast  train ;  for  the  resistance  of  the  air,  which  is  a  small  part 
of  the  total  resistance  so  long  as  the  train  is  moving  slowly,  becomes 
the  predominant  factor  at  high  speeds.  The  resistance  offered  to  steam- 
boats running  at  high  speeds  is  usually  considered  to  increase  as  the 
cube  of  the  velocity.  Thus  the  Cedric,  of  the  White  Star  Line,  having 
a  speed  of  17  knots,  has  a  horse  power  of  14,000,  and  a  total  weight 
when  loaded  of  about  38,000  tons,  while  the  Kaiser  Wilhelm  II,  of  the 
North  German  Lloyd  Line,  having  a  speed  of  24  knots,  has  engines 
of  40,000  horse  power,  although  the  total  weight  when  loaded  is  only 
26,000  tons. 

QUESTIONS  AND  PROBLEMS 

1.  In  what  respects  is  friction  an  advantage,  and  in  what  a  dis- 
advantage, in  everyday  life?  Could  we  get  along  without  it? 

2.  Why  is  a  stream  swifter  at  the  center  than  at  the  banks? 

3.  A  smooth  block  is  10  x  8  x  3  in.  Compare  the  distances  which  it  will 
slide  when  given  a  certain  initial  velocity  on  smooth  ice,  if  resting  first  on 
a  10  x  8  face ;  second,  on  a  10  X  3  face ;  and  third,  on  an  8  x  3  face. 

4.  Why  is  sand  often  placed  on  a  track  in  starting  a  heavy  train? 

5.  What  is  the  coefficient  of  friction  of -brass  on  brass  if  a  force  of 
25  Ib.  is  required  to  maintain  uniform  motion  in  a  brass  block  weighing 
200  Ib.,  when  it  slides  horizontally  on  a  brass  bed  ? 

6.  Why  does  a  team  have  to  keep  pulling  after  a  load  is  started  ? 


156  WORK  AND  HEAT  ENERGY 

7.  The  coefficient  of  friction  between  a  block  and  a  table  is  .3. 
What  force  will  be  required  to  keep  a  500-g.  block  in  uniform  motion  ? 

8.  In  what  way  is  friction  an  advantage  in  lifting  buildings  with  a 
jackscrew  ?   In  what  way  is  it  a  disadvantage  ? 


EFFICIENCY 

202.  Definition  of  efficiency.     Since  it  is  only  in  an  ideal 
machine  that  there  is  no  friction,  in  all  actual  machines  the 
work  done  by  the  acting  force  always  exceeds,  by  the  amount 
of  the  work  done  against  friction,  the  amount  of  potential 
and  kinetic  energy  stored  up.    We  have  seen  that  the  former 
is  wasted  work  in  the  sense  that  it  can  never  be  regained. 
Since  the  energy  stored  up  represents  work  which  can  be 
regained,  it  is  termed  useful  work.    In  most  machines  an  effort 
is   made  to  have  the  useful  work    as    large    a  fraction   of 
the  total  work  expended  as  possible.    The  ratio  of  the  use/id 
work  to  the  total  work  done  by  the  acting  force  is  called  the 
EFFICIENCY  of  the  machine.    Thus 

^nn  .  Useful  work  accomplished 

Efficiency  =  — — — —  r,    . (1) 

Total  work  expended 

Thus,  if  in  the  system  of  pulleys  shown  in  Fig.  138  it  is  necessary  to 
add  a  weight  of  50  g.  at  jp  in  order  to  pull  up  slowly  an  added  weight  of 
240  g.  at  W,  the  work  done  by  the  50  g.  while  F  is  moving  over  1  cm. 
will  be  50  x  1  g.  cm.  The  useful  work  accomplished  in  the  same  time 

240  x  i 

is  240  x  ^  g.  cm.    Hence  the  efficiency  is  equal  to  — *•  =  f  =  80%. 

50  x  1 

203.  Efficiencies  of  some  simple  machines.    In  simple  levers 
the  friction  is  generally  so  small  as  to  be  negligible  ;  hence  the 
efficienc}r  of  such  machines  is  approximately  100%.    When 
inclined  planes  are  used  as  machines  the  friction  is  also  small, 
so  that  the  efficiency  generally  lies  between  90%  and  100%. 
The  efficiency  of  the  commercial  block  and  tackle  (Fig.  138), 
with  several  movable  pulleys,  is  usually  considerably  less, 
varying  between  40%   and  60%.    In  the  jackscrew  there  is 


EFFICIENCY 


15T 


necessarily  a  very  large  amount  of  friction,  so  that  although 
the  mechanical  advantage  is  enormous,  the  efficiency  is  often 
as  low  as  25%.  The  differential  pulley  of  Fig.  159  has  also  a 
very  high  mechanical  advantage  with  a  very  small  efficiency. 
Gear  wheels  such  as  those  shown 
in  Fig.  157,  or  chain  gears  such  as 
those  used  in  bicycles,  are  machines 
of  comparatively  high  efficiency,  often 
utilizing  between  90%  and  100%  of 
the  energy  expended  upon  them. 


FIG.  172.    Overshot  water 
wheel 


204.  Efficiency  of  overshot  water  wheels. 

The  overshot  water  wheel  (Fig.  172)  utilizes 
chiefly  the  potential  energy  of  the  water  at 
S ;  for  the  wheel  is  turned  by  the  weight 
of  the  water  in  the  buckets.  The  work  ex- 
pended on  the  wheel  per  second,  in  foot 
pounds  or  gram  centimeters,  is  the  product 
of  the  weight  of  the  water  which  passes  over 
it  per  second  by  the  distance  through  which 

it  falls.  The  efficiency  is  the  work  which  the  wheel  can  accomplish 
in  a  second,  divided  by  this  quantity.  Such  wheels  are  very  common  in 
mountainous  regions,  where  it  is  easy  to  obtain  considerable  fall,  but 
where  the  streams  carry  a  small  volume  of  water.  The  efficiency  is  high, 
being  often  between  80%  and  90%.  The  loss  is  due>not  only  to  the  friction 
in  the  bearings  and  gears  (see  C),  but  also 
to  the  fact  that  some  of  the  water  is  spilled 
from  the  buckets,  or  passes  over  without 
entering  them  at  all.  This  may  still  be  re- 
garded as  a  frictional  loss,  since  the  energy 
disappears  in  internal  friction  when  the 
water  strikes  the  ground. 

205.  Efficiency  of  undershot  water  wheels. 
The  old-style  undershot  wheel  (Fig.  173), 
so  common  in  flat  countries  where  there 

is  little  fall  but  abundance  of  water,  utilizes  only  the  kinetic  energy 
of  the  water  running  through  the  race  from  A.  It  seldom  transforms 
into  useful  work  more  than  25%  or  30%  of  the  potential  energy  of  the 
water  above  the  dam.  There  are,  however,  certain  modern  forms  of 


FIG.  173.   The  undershot 
wheel 


158 


WORK  AND  HEAT  ENERGY 


undershot  wheel  which  are  extremely  efficient.  For  example,  the  Pelton 
ivheel  (Fig.  174),  developed  since  1880,  and  now  very  commonly  used  for 
small-power  purposes  in  cities  supplied  with  water- 
works, sometimes  has  an  efficiency  as  high  as  83%. 
The  water  is  delivered  from  a  nozzle  0  against 
cup-shaped  buckets  arranged  as  in  the  figure. 

206.  Efficiency  of  water  turbines.  The  turbine 
wheel  was  invented  in  France  in  1833,  and  is  now 
used  more  than  any  other  form  of  water  wheel. 
It  stands  completely  under  water  in  a  case  at 
the  bottom  of  a  turbine  pit,  rotating  in  a  hori- 
zontal plane.  Fig.  175  shows  the  method  of  in- 
stalling a  turbine  at  Niagara.  C  is  the  outer  case  into  which  the 

water  enters  from  the  penstock  p.  Fig.  176, 
(1),  shows  the  outer  case  with  contained 


FIG.  174.   The  Pelton 
water  motor 


FIG.  175.   A  turbine 
installed 


FIG.  176.   The  turbine  wheel 

(1)  Outer  case ;    (2)  inner  case ;    (3)  rotating 
part;    (4)  section 

turbine ;  Fig.  176,  (2),  is  the  inner  case  in  which  are  the  fixed  guides  G, 
which  direct  the  water  at  the  most  advantageous  angle  against  the  blades 


MECHANICAL  EQUIVALENT  OF  HEAT          159 

of  the  wheel  inside  ;  Fig.  176,  (3),  is  the  wheel  itself ;  and  Fig.  176,  (4), 
is  a  section  of  wheel  and  inner  case,  showing  how  the  water  enters 
through  the  guides  and  impinges  upon  the  blades  W.  The  spent  water 
simply  falls  down  from  the  blades  into  the  tailrace  T  (Fig.  175).  The 
amount  of  water  which  passes  through  the  turbine  can  be  controlled 
by  means  of  the  rod  P  [Fig.  176,  (1)],  which  can  be  turned  so  as  to 
increase  or  decrease  the  size  of  the  openings  between  the  guides  G 
[Fig.  176,  (2)].  The  energy  expended  upon  the  turbine  per  second  is 
the  product  of  the  mass  of  water  which  passes  through  it  by  the  height 
of  the  turbine  pit.  Efficiencies  as  high  as  90%  have  been  attained  with 
such  wheels.  One  of  the  most  powerful  turbines  in  existence  is  at 
Shawenegan  Falls,  Quebec,  Canada.  The  pit  is  135  feet  deep,  the  wheel 
10  feet  in  diameter,  and  the  horse  power  developed  10,500. 

QUESTIONS  AND  PROBLEMS 

1.  If  it  is  necessary  to  pull  on  a  block  and  tackle  with  a  force  of  100 
Ib.  in  order  to  lift  a  weight  of  300  lb.,  and  if  the  force  must  move  6  ft. 
to  raise  the  weight  1  ft.,  what  is  the  efficiency  of  the  system  ? 

2.  If  the  efficiency  had  been  65%,  what  force  would  have  been  neces- 
sary in  the  preceding  problem  ? 

3.  The  largest  overshot  water  wheel  in  existence  is  at  Laxey,  on  the 
Isle  of  Man.    It  has  a  horse  power  of  150,  a  diameter  of  72.5  ft.,  and  an 
efficiency  of  85%.    How  many  cubic  feet  of  water  pass  over  it  per  second  ? 

4.  The  Niagara  turbine  pits  are  136  ft.  deep  and  their  average  horse 
power  is  5000.    Their  efficiency  is  85%.    How  much  water  does  each 
turbine  discharge  per  minute  ? 

5.  There  is  a  Pelton  wheel  at  the  Sutro  tunnel  in  Nevada  which  is 
driven  by  water  supplied  from  a  reservoir  2100  ft.  above  the  level  of  the 
motor.    The  diameter  of  the  nozzle  is  about  ^  in.,  and  that  of  the  wheel 
but  3  ft.,  yet  100  H.P.  is  developed.   If  the  efficiency  is  80%,  how  many 
cubic  feet  of  water  are  discharged  per  second? 

MECHANICAL  EQUIVALENT  OF  HEAT  * 

207.  What  becomes  of  wasted  work?  In  all  of  the  devices 
for  transforming  work  which  we  have  considered  we  have 
found  that  on  account  of  frictional  resistances  a  certain  per 
cent  of  the  work  expended  upon  the  machine  is  wasted.  The 

*  This  subject  should  be  preceded  by  a  laboratory  experiment  upon  the  "law 
of  mixtures,"  and  either  preceded  or  accompanied  by  experiments  upon  specific 
heat  and  mechanical  equivalent.  See  authors'  manual,  Experiments  18, 19,  and  20. 


160  WORK  AND  HEAT  ENERGY 

question  which  at  once  suggests  itself  is,  "  What  becomes  of 
this  wasted  work  ?  "  The  following  familiar  facts  suggest  an 
answer.  When  two  sticks  are  vigorously  rubbed  together  they 
become  hot ;  augers  and  drills  often  become  too  hot  to  hold ; 
matches  are  ignited  by  friction ;  if  a  strip  of  lead  be  struck  a 
few  sharp  blows  with  a  hammer,  it  is  appreciably  warmed. 
Now  since  we  learned  in  Chapter  IV  that,  according  to  modern 
notions,  increasing  the  temperature  of  a  body  means  simply 
increasing  the  average  velocity  of  its  molecules,  and  therefore 
their  average  kinetic  energy,  the  above  facts  point  strongly  to 
the  conclusion  that  in  each  case  the  mechanical  energy  ex- 
pended has  been  simply  transformed  into  the  energy  of  molec- 
ular motion.  This  view  was  first  brought  into  prominence  in 
1798  by  Benjamin  Thompson,  Count  Rumford,  an  American 
by  birth,  who  was  led  to  it  by  observing  that  in  the  boring  of 
cannon  heat  was  continuously  developed.  It  was  first  care- 
fully tested  by  the  English  physicist,  James  Prescott  Joule 
(1818-1889),  in  a  series  of  epoch-making  experiments  extend- 
ing from  1842  to  1870.  In  order  to  understand  these  experi- 
ments we  must  first  learn  how  heat  quantities  are  measured. 
208.  Unit  of  heat  —  the  calorie.  A  unit  of  heat  is  defined 
as  the  amount  of.  heat  which  is  required  to  raise  the  temperature 
of  1  gram  of  water  through  1°  centigrade.  This  unit  is  called 
the  calorie.  Thus,  for  example,  when  a  hundred  grams  of  water 
has  its  temperature  raised  four  degrees,  we  say  that  four  hun- 
dred calories  of  heat  have  entered  the  water.  Similarly,  when 
a  hundred  grams  of  water  has  its  temperature  lowered  ten 
degrees,  we  say  that  a  thousand  calories  have  passed  out  of 
the  water.  If,  then,  we  wish  to  measure,  for  instance,  the 
amount  of  heat  developed  in  a  lead  bullet  when  it  strikes 
against  a  target,  we  have  only  to  let  the  spent  bullet  fall  into 
a  known  weight  of  water  and  to  measure  the  number  of 
degrees  through  which  the  temperature  of  the  water  rises. 
The  product  of  the  number  of  grams  of  water  by  its  rise  in 


COUNT  RUMFORD  (BENJAMIN  THOMPSON,  1753-1814) 

"  An  eminent  scientist,  enlightened  philanthropist,  and  sagacious 
public  administrator,"  was  born  at  Woburn,  Massachusetts;  was 
a  Tory  during  the  Revolution;  removed  to  England,  and  after- 
ward to  Munich,  where  he  lived  for  eleven  years  as  minister  of 
war,  minister  of  police,  and  grand  chamberlain  to  the  elector. 
He  reorganized  the  Bavarian  army,  suppressed  beggary,  provided 
employment  for  the  poor,  and  established  industrial  schools.  He 
was  one  of  the  earliest  and  most  influential  advocates  of  the  view 
that  heat  is  a  mode  of  molecular  motion.  He  invented  the 
Rumford  photometer  (see  §  445,  p.  367) 


MECHANICAL  EQUIVALENT  OF  HEAT          161 

temperature  is  then,  by  definition,  the  number  of  calories  of 
heat  which  have  passed  into  the  water. 

It  will  be  noticed  that  in  the  above  definition  we  make  no 
assumption  whatever  as  to  what  heat  is.  Previous  to  the  nine- 
teenth century  physicists  generally  held  it  to  be  an  invisible, 
weightless  fluid,  the  passage  of  which  into  or  out  of  a  body 
caused  it  to  grow  hot  or  cold.  This  view  accounts  well  enough 
for  the  heating  which  a  body  experiences  when  it  is  held  in 
contact  with  a  flame  or  other  hot  body,  but  it  has  difficulty 
in  explaining  the  heating  produced  by  rubbing  or  pounding. 
Rumf ord's  view  accounts  easily  for  this,  as  we  have  seen,  while 
it  accounts  no  less  easily  for  the  heating  of  cold  bodies  by  con- 
tact with  hot  ones  ;  for  we  have  only  to  think  of  the  hotter  and 
therefore  more  energetic  molecules  of  the  hot  body  as  com- 
municating their  energy  to  the  molecules  of  the  colder  body 
in  much  the  same  way  in  which  a  rapidly  moving  billiard  ball 
transfers  part  of  its  kinetic  energy  to  a  more  slowly  moving 
ball  against  which  it  strikes. 

209.  Joule's  experiment  on  the  heat  developed  by  friction. 
Joule  argued  that  if  the  heat  produced  by  friction,  etc.  is 
indeed  merely  mechanical  energy  which  has  been  transferred 
to  the  molecules  of  the  heated  body,  then  the  same  number 
of  calories  must  always  be  produced  by  the  disappearance  of 
a  given  amount  of  mechanical  energy.  And  this  must  be 
true,  no  matter  whether  the  work  is  expended  in  overcoming 
the  friction  of  wood  on  wood,  of  iron  on  iron,  in  percussion, 
in  compression,  or  in  any  other  conceivable  way.  To  see 
whether  or  not  this  were  so,  he  caused  mechanical  energy  to 
disappear  in  as  many  ways  as  possible  and  measured  in  every 
case  the  amount  of  heat  developed. 

In  his  first  experiment  he  caused  paddle  wheels  to  rotate  in  a  vessel 
of  water  by  means  of  falling  weights  W  (Fig.  177).  The  amount  of 
work  done  by  gravity  upon  the  weights  in  causing  them  to  descend 
through  any  distance  d  was  equal  to  their  weight  W  times  this  distance. 


162 


WOBK  AND    HEAT    ENERGY 


If  the  weights  descended  slowly  and  uniformly,  this  work  was  all 
expended  in  overcoming  the  resistance  of  the  water  to  the  motion  of 
the  paddle  wheels  through  it ;  that  is,  it  was  wasted  in  eddy  currents 
in  the  water.  Joule  measured 
the  rise  in  the  temperature  of 
the  water  and  found  that  the 
mean  of  his  three  best  trials 
gave  427  gram  meters  as  the 
amount  of  work  required  to  de- 
velop enough  heat  to  raise  a  gram 
of  water  one  degree.  He  then 
repeated  the  experiment,  substi- 
tuting mercury  for  water,  and 
obtained  425  gram  meters  as  the 
work  necessary  to  produce  a  cal- 
orie of  heat.  The  difference  be- 


FIG.  177.   Joule's  first  experiment  on 
the  mechanical  equivalent  of  heat 


tween  these  numbers  is  less  than  was  to  have  been  expected  from  the 
unavoidable  errors  in  the  observations.  He  then  devised  an  arrange- 
ment in  which  the  heat  was  developed  by  the  friction  of  iron  on  iron, 
and  again  obtained  425. 

210.  Heat  produced  by  collision.  A  Frenchman  by  the  name 
of  Hirn  was  the  first  to  make  a  careful  determination  of  the 
relation  between  the  heat  developed  by  collision  and  the  kinetic 
energy  which  disappears.    He  allowed  a  steel  cylinder  to  fall 
through  a  known  height  and  crush  a  lead  ball  by  its  impact 
upon  it.   The  amount  of  heat  developed  in  the  lead  was  meas- 
ured by  observing  the  rise  in  temperature  of  a  small  amount  of 
water  into  which  the  lead  was  quickly  plunged.   As  the  mean 
of  a  large  number  of  trials  he  also  found  that  425  gram  meters 
of  energy  disappeared  for  each  calorie  of  heat  which  appeared. 

211.  Heat  produced  by  the  compression  of  a  gas.   Another 
way  in  which  Joule  measured  the  relation  between  heat  and 
work  was  by  compressing  a  gas  and  comparing  the  amount 
of  work  done  in  the  compression  with  the  amount  of  heat 
developed. 

Every  bicyclist  is  aware  of  the  fact  that  when  he  inflates  his 
tires  the  pump  grows  hot.    This  is  due  partly  to  the  friction  of 


MECHANICAL  EQUIVALENT  OF  HEAT          163 

the  piston  against  the  walls,  but  chiefly  to  the  fact  that  the 
downward  motion  of  the  piston  is  transferred  to  the  molecules 
which  come  in  contact  with  it,  so  that  the  velocity  of  these 
molecules  is  increased.  The  principle  is  precisely  the  same 
as  that  involved  in  the  velocity  communicated  to  a  ball  by  a 
bat.  If  the  bat  is  held  rigidly  fixed  and  a  ball  thrown  against 
it,  the  ball  rebounds  with  a  certain  velocity ;  but  if  the  bat 
is  moving  rapidly  forward  to  meet  the  ball,  the 
latter  rebounds  with  a  much  greater  velocity. 
So  the  molecules  which  in  their  natural  motions 
collide  with  an  advancing  piston,  rebound  with 
greater  velocity  than  they  would  if  they  had 
impinged  upon  a  fixed  wall.  This  increase  in 
the  molecular  velocity  of  a  gas  on  compression 
is  so  great  that  when  a  mass  of  gas  at  0°  centi- 
grade is  compressed  to  one  half  its  volume  the 
temperature  rises  to  87°  centigrade. 


The  effect  may  be  strikingly  illustrated  by  the  fire 
syringe  (Fig.  178).    Let  a  few  drops  of  carbon  bisul-     -^       -jyg    rp^ 
phide  be  placed  on  a  small  bit  of  cotton,  dropped  to  the       gre  Syrino-e 
bottom  of  the  tube  A,  and  then  removed;  then  let  the 
piston  B  be  inserted  and  very  suddenly  depressed.    Sufficient  heat  will 
be  developed  to  ignite  the  vapor  and  a  flash  wyi  result.    (If  the  flash 
does  not  result  from  the  first  stroke,  withdraw  the  piston  completely, 
then  reinsert,  and  compress  again.) 

To  measure  the  heat  of  compression  Joule  surrounded  a 
small  compression  pump  with  water,  took  300  strokes  on  the 
pump,  and  measured  the  rise  in  temperature  of  the  water.  As 
the  result  of  these  measurements  he  obtained  444  gram  meters 
as  the  "  mechanical  equivalent"  of  the  calorie.  The  experiment, 
however,  could  not  be  performed  with  great  exactness. 

212.  Cooling  by  expansion.  Joule  also  obtained  the  rela- 
tion between  heat  and  work  from  experiments  on  the  cooling 
produced  by  expansion.  This  process  is  exactly  the  converse 


164  WOEK  AND  HEAT  ENERGY 

of  heating  by  compression.  If  a  compressed  gas  is  allowed  to 
expand  and  force  out  a  piston,  or  merely  to  expand  against 
atmospheric  pressure,  it  is  always  found  to  be  cooled  by  the 
process.  This  is  because  the  kinetic  energy  of  the  molecules 
is  transferred  to  the  piston,  so  that  the  former  rebound  from 
the  latter  with  less  velocity  than  they  had  before  the  blow. 
The  refrigerators  used  on  shipboard  are  good  illustrations  of 
this  principle.  Air  is  compressed  by  an  engine  to  perhaps 
one  fourth  its  natural  volume.  The  heat  generated  by  the 
compression  is  then  removed  by  causing  the  air  to  circulate 
about  pipes  kept  cool  by  the  flow  of  cold  water  through  them. 
This  compressed  air  is  then  allowed  to  expand  into  the  refrig- 
erating chamber,  the  temperature  of  which  is  thus  lowered 
many  degrees. 

Joule's  determination  of  the  mechanical  equivalent  of  heat 
from  the  amount  of  work  done  by  an  expanding  gas  and  the 
amount  of  heat  lost  in  expansion  gave  '437  gram  meters. 
This  experiment  also  was  one  for  which  no  great  amount  of 
exactness  could  be  claimed. 

213.  Significance  of  Joule's  experiments.  Joule  made  three 
other  determinations  of  the  relation  between  heat  and  work 
by  methods  involving  electrical  measurements.  He  published 
as  the  mean  of  all  his  determinations  426.4  gram  meters  as 
the  mechanical  equivalent  of  the  calorie.  But  the  value 
of  his  experiments  does  not  lie  primarily  in  the  accuracy  of 
the  final  results,  but  rather  in  the  proof  which  they  for  the 
first  time  furnished  that  whenever  a  given  amount  of  work  is 
wasted,  no  matter  in  what  particular  way  this  waste  may  occur, 
there  is  always  an  appearance  of  the  same  definite  invariable 
amount  of  heat. 

The  most  accurate  determination  of  the  mechanical  equiva- 
lent of  heat  was  made  by  Rowland  (1848-1901)  in  1880 
at  Johns  Hopkins  University.  He  obtained  427  gram  meters 
(4.19  xlO7 ergs).  We  shall  generally  take  it  as  42,000,000  ergs. 


MECHANICAL  EQUIVALENT  OF  HEAT  165 

214.  The  conservation  of  energy.    We  are  now  in  a  position 
to  state  the  law  of  all  machines  in  its  most  general  form ;  that 
is,  in  such  a  way  as  to  include  even  the  cases  where  friction 
is  present.    It  is :    The  work  done  by  the  acting  force  is  equal  to 
the  sum  of  the  kinetic  and  potential  energies  stored  up  plus  the 
mechanical  equivalent  of  the  heat  developed. 

In  other  words,  whenever  energy  is  expended  on  a  machine  or 
device  of  any  kind,  an  exactly  equal  amount  of  energy  always 
appears  either  as  useful  work  or  as  heat.  The  useful  work  may 
be  represented  in  the  potential  energy  of  a  lifted  mass,  as 
when  water  is  pumped  up  to  a  reservoir ;  or  in  the  kinetic 
energy  of  a  moving  mass,  as  when  a  stone  is  thrown  from  a 
sling ;  or  in  the  potential  energies  of  molecules  whose  posi- 
tions with  reference  to  one  another  have  been  changed,  as 
when  a  spring  has  been  bent;  or  in  the  molecular  potential 
energy  of  chemically  separated  atoms,  as  when  an  electric 
current  separates  a  compound  substance.  The  wasted  work 
always  appears  in  the  form  of  increased  molecular  motion; 
that  is,  in  the  form  of  heat.  This  important  generalization 
has  received  the  name  of  the  Principle  of  the  Conservation  of 
Energy.  It  may  be  stated  thus :  Energy  may  be  transformed, 
but  it  can  never  be  created  or  destroyed. 

215.  Perpetual  motion.    In  all  ages  tlifere  have  been  men 
who  have  spent  their  lives  in  trying  to  invent  a  machine  out 
of  which  work  could  be  continually  obtained,  without  the  ex- 
penditure of  an  equivalent  amount  of  work  upon  it.    Such 
devices  are  called  perpetual-motion  machines.    Even  to  this 
day  the  United  States  patent  office  annually  receives  scores 
of  applications  for  patents  on  such  devices.    The  possibility 
of  the  existence  of  such  a  device  is  absolutely  denied  by  the 
statement  of  the  principle  of  the  conservation  of  energy.    For 
only  in  case  there  is  no  heat  developed,  that  is,  in  case  there 
are  no  frictional  losses,  can  the  work  taken  out  be  equal  to 
the  work  put  in,  and  in  no  case  can  it  be  greater.    Since,  in 


166  WORK  AND  HEAT  ENERGY 

fact,  there  are  always  some  frictional  losses,  the  principle  of  the 
conservation  of  energy  asserts  that  it  is  impossible  to  make  a 
machine  which  will  keep  itself  running  forever,  even  though  it 
does  no  useful  work  ;  for  no  matter  how  much  kinetic  or  poten- 
tial energy  is  imparted  to  the  machine  to  begin  with,  there 
must  always  be  a  continual  drain  upon  this  energy  to  overcome 
frictional  resistances  ?  so  that,  as  soon  as  the  wasted  work  has 
become  equal  to  the  initial  energy,  the  machine  must  stop. 

The  first  man  to  make  a  formal  and  complete  statement  of 
the  principle  of  the  conservation  of  energy  was  the  German 
physician  Robert  Mayer,  whose  work  was  published  in  1842. 
Twenty  years  later,  partly  through  the  theoretical  writings 
of  Helmholtz  and  Clausius  in  Germany,  and  of  Kelvin  and 
Rankine  in  England,  but  more  especially  through  the  experi- 
mental work  of  Joule,  the  principle  had  gained  universal 
recognition  and  had  taken  the  place  which  it  now  holds  as 
the  corner  stone  of  all  physical  science. 

216.  Transformations  of  energy  in  a  power  plant.  The  transforma- 
tions of  energy  which  take  place  in  any  power  plant,  such  as  that  at 
Niagara,  are  as  follows :  The  energy  first  exists  as  the  potential  energy 
of  the  water  at  the  top  of  the  falls.  This  is  transformed  in  the  turbine 
pits  into  the  kinetic  energy  of  the  rotating  wheels.  These  turbines 
drive  dynamos  in  which  there  is  a  transformation  into  the  energy  of 
electric  currents.  These  currents  travel  on  wires  as  far  as  Syracuse, 
150  miles  away,  where  they  run  street  cars  and  other  forms  of  motors. 
The  principle  of  conservation  of  energy  asserts  that  the  work  which 
gravity  did  upon  the  water  in  causing  it  to  descend  from  the  top  to  the 
bottom  of  the  turbine  pits  is  exactly  equal  to  the  work  done  by  all  the 
motors,  plus  the  heat  developed  in  all  the  wires  and  bearings,  and  in 
the  eddy  currents  in  the  water: 

Let  us  next  consider  where  the  water  at  the  top  of  the  falls  obtained 
its  potential  energy.  •  Water  is  being  continually  evaporated  at  the  sur- 
face of  the  ocean  by  the  sun's  heat.  This  heat  imparts  sufficient  kinetic 
energy  to  the  molecules  to  enable  them  to  break  away  from  the  attrac- 
tions of  their  fellows  and  to  rise  above  the  surface  in  the  form  of  vapor. 
The  lifted  vapor  is  carried  by  winds  over  the  continents  and  precipitated 
in  the  form  of  rain  or  snow.  Thus  the  potential  energy  of  the  water 


SPECIFIC  HEAT  16T 

above  the  falls  at  Niagara  is  simply  transformed  heat  energy,  of  the 
sun.  If,  in  this  way,  we  analyze  any  available  source  of  energy  at 
man's  disposal,  we  find  in  practically  every  case  that  it  is  directly  trace- 
able to  the  sun's  heat  as  its  source.  Thus  the  energy  contained  in  coal 
is  simply  the  energy  of  separation  of  the  oxygen  and  carbon  which  were 
separated  in  the  processes  of  growth.  This  separation  was  effected  by 
the  sun's  rays. 

We  can  form  some  conception  of  the  enormous  amount  of  energy 
which  the  sun  radiates  in  the  form  of  heat  by  reflecting  that  of  this  heat 
the  earth  receives  not  more  than  o,000  0100  000  part.  Of  the  amount 
received  by  the  earth  not  more  than  10100  part  is  stored  up  in  animal 
and  vegetable  life  and  lifted  water.  This  is  practically  all  of  the  energy 
which  is  available  on  the  earth  for  man's  use. 


QUESTIONS  AND  PROBLEMS 

1.  Analyze  the  transformations  of  energy  which  occur  when  a  bullet 
is  fired  vertically  upward. 

2.  Show  that  the  energy  of  a  water  fall  is  merely  transformed  solar 
energy. 

3.  How  many  calories  of  heat  are  generated  by  the  impact  of  a 
200-kilo  bowlder  when  it  falls  vertically  through  100  m.  ? 

4.  Thousands  of  meteorites  are  falling  into  the  sun  with  enormous 
velocities  every  minute.    From  a  consideration  of  the  preceding  example 
account  for  a  portion,  at  least,  of  the  sun's  heat. 

5.  The  Niagara  Falls  are  160  ft.  high.    How  much  warmer  is  the 
water  at  the  bottom  of  the  falls  than  at  the  top  ? 

6.  Why  does  air  escaping  from  a  bicycle  tire  feel  cold  ? 

SPECIFIC  HEAT 

21.7.  Definition  of  specific  heat.  When  we  experiment  upon 
different  substances  we  find  that  it  requires  wholly  different 
amounts  of  heat  energy  to  produce  in  one  gram  of  mass  one 
degree  of  change  in  temperature. 

Let  100  g.  of  lead  shot  be  'placed  in  one  test  tube,  100  g.  of  bits  of 
iron  wire  in  another,  and  100  g.  of  aluminium  wire  in  a  third.  Let 
them  all  be  placed  in  a  pail  of  boiling  water  for  ten  or  fifteen  minutes, 
care  being  taken  not  to  allow  any  of  the  water  to  enter  any  of  the  tubes. 
Let  three  small  vessels  be  provided,  each  of  which  contains  100  g.  of 


168  WOKK  AND  HEAT  ENERGY 

water  at  the  temperature  of  the  room.  Let  the  heated  shot  be  poured 
into  the  first  beaker,  and  after  thorough  stirring  let  the  rise  in  the 
temperature  of  the  water  be  noted.  Let  the  same  be  done  with  the 
other  metals.  The  aluminium  will  be  found  to  raise  the  temperature 
about  twice  as  much  as  the  iron,  and  the  iron  about  three  times  as 
much  as  the  lead.  Hence,  since  the  three  metals  have  cooled  through 
approximately  the  same  number  of  degrees,  we  must  conclude  that 
about  six  times  as  much  heat  has  passed  out  of  the  aluminium  as  out 
of  the  lead ;  that  is,  each  gram  of  aluminium  in  cooling  1°  C.  gives  out 
about  six  times  as  many  calories  as  a  gram  of  lead. 

The  number  of  calories  taken  up  by  1  gram  of  a  substance 
when  its  temperature  rises  through  1°  C.,  or  given  up  when  it 
falls  through  1°  (7.,  is  called  the  specific  heat  of  that  substance. 

It  will  be  seen  from  this  definition,  and  the  definition  of  the 
calorie,  that  the  specific  heat  of  water  is  1. 

218.  Determination  of  specific  heat  by  the  method  of  mix- 
tures. The  preceding  experiments  illustrate  a  method  for 
measuring  accurately  the  specific  heats  of  different  substances. 
For,  in  accordance  with  the  principle  of  the  conservation  of 
energy,  when  hot  and  cold  bodies  are  mixed,  as  in  these  ex- 
periments, so  that  heat  energy  passes  from  one  to  the  other, 
the  gain  in  the  heat  energy  of  one  must  be  just  equal  to  the  loss 
in  the  heat  energy  of  the  other. 

This  method  is  by  far  the  most  common  one  for  determin- 
ing the  Specific  heats  of  substances.  It  is  known  as  the  method 
of  mixtures. 

Suppose,  to  take  an  actual  case,  that  the  initial  'temperature  of  the 
shot  used  in  §  217  was  95°  C.,  and  that  of  the  water  19.7°,  and  that,  after 
mixing,  the  temperature  of  the  water  and  shot  was  22°.  Then,  since 
100  g.  of  water  has  had  its  temperature  raised  through  22°  —  19.7°  =  2.3°, 
we  know  that  230  calories  of  heat  have  entered  the  water.  Since  the 
temperature  of  the  shot  fell  through  95°  —  22°  =  73°,  the  number  of 
calories  given  up  by  the  100  g.  of  shot  in  falling  1°  was  2-j*^.  —  3.15. 
Hence  the  spjacjficjieat  of  lead,  that  is,  the  number  of  calories  of  heat  given 

3  15 

up  by  1  gram  of  lead  when  its  temperature  falls  1°  C.  is  -—  =  .0315. 


SPECIFIC  HEAT  169 

Or  again,  we  may  work  out  the  problem  algebraically  as  follows : 
Let  x  equal  the  specific  heat  of  lead.  Then  the  number  of  calories  which 
come  out  of  the  shot  is  its  mass  times  its  specific  heat  times  its  change  in 
temperature  ;  that'is,  100  x  a:  x  (95  —  22),  and,  similarly,  the  number  which 
enter  the  water  is  the  same,  namely  100  x  1  x  (22  —  19.7).  Hence  we  have 

100  (95  -  22)  x  =  100  (22  -  19.7)     or     x  =  .0315. 

By  experiments  of  this  sort  the  specific  heats  of  some  of  the 
common  substances  have  been  found  to  be  as  follows : 

TABLE  OF  SPECIFIC  HEATS 

Aluminium 218  Iron .    .    .113 

Brass 094  Lead 0315 

Copper       095  Mercury 0333 

Glass      2  Platinum 032 

Gold       .~T 0316  Silver 0568 

Ice  .504  Zinc 0935 


QUESTIONS   AND   PROBLEMS 

1.  Why  is  a  liter  of  hot  water  a  better  foot  warmer  than  an  equal 
volume  of  any  substance  in  the  preceding  table  ? 

2.  Which  would  be  heated  more,  a  lead  or  a  steel  bullet,  if  they  were 
fired  against  a  target  with  equal  speeds  ? 

3.  If  100  g.  of  mercury  at  95°  C.  are  mixed  with  100  g.  of  water  at 
15° C.,  and  if  the  resulting  temperature  is  17. 6° C.,  what  is  the  specific 
heat  of  mercury  ? 

4.  A  10-g.  bullet  of  lead  is  shot  from  a  gun  with  a  velocity  of  400  m. 
per  second.   Through  how  many  degrees  centigrade  is  its  temperature 
raised  when  it  strikes  a  target?    (Assume  that  all  of  the  heat  stays  in 
the  bullet.) 

5.  From  what  height  must  a  block  of  lead  fall  in  order  to  have  its 
temperature  raised  through  1°  C.  ? 

6.  If  200  g.  of  water  at  80°  C.  are  mixed  with  100  g.  of  water  at 
10° C.,  what  will  be  the  temperature  of  the  mixture?     (Let  x  equal  the 
final  temperature ;  then  100  (x  —  10)  calories  are  gained  by  the  cold 
water,  while  200  (80  —  x)  calories  are  lost  by  the  hot  water.) 

7.  What  temperature  will  result  if  300  g.  of  copper  at  100°  C.  are 
placed  in  200  g.  of  water  at  15°  C.  ? 

8.  The  specific  heat  of  water  is  much  greater  than  that  of  any 
other  liquid  or  of  any  solid.    Explain  how  this  accounts  for  the  fact 


170  WOBR  AND  HEAT  ENERGY 

that  an  island  in  mid-ocean  undergoes  less  extremes  of  temperature 
than  an  inland  region. 

9.  How  many  grams  of  ice-cold  water  must  be  poured  into  a  tum- 
bler weighing  300  g.,  to  cool  it  from  60°  C.  to  20°  C.,  -the  specific  heat 
of  glass  being  .2  ? 

10.    If  you  put  a  20  g.  silver  spoon  at  20°  C.  into  a  150-cc.  cup  of  tea 
at  70° C.,  how  much  do  you  cool  the  tea? 


CHANGE  OF  STATE  —  FUSION* 

219.  Heat  of  fusion.  If  on  a  cold  day  in  winter  a  quantity  of  snow 
ts  brought  in  from  out  of  doors,  where  the  temperature  is  below  0°C., 
and  placed  over  a  source  of  heat,  a  thermometer  plunged  into  the  snow 
will  be  found  to  rise  slowly  until  the  temperature  reaches  0°  C.,  when  it 
will  become  stationary  and  remain  so  during  all  the  time  that  the  snow 
is  melting,  provided  only  that  the  contents  of  the  vessel  are  continu- 
ously and  vigorously  stirred.  As  soon  as  the  snow  is  all  melted  the 
temperature  will  begin  to  rise  again. 

Since  the  temperature  of  ice  at  0°  C.  is  the  same  as  the 
temperature  of  water  at  0°  C. ,  it  is  evident  from  this  experiment 
that  when  ice  is  being  changed  to  water,  the  entrance  of  heat 
energy  into  it  does  not  produce  any  change  in  the  average 
kinetic  energy  of  its  molecules.  This  energy  must  therefore 
all  be  expended  in  pulling  apart  the  molecules  of  the  crystals 
of  which  the  ice  is  composed  and  thus  reducing  it  to  a  form 
in  which  the  molecules  are  held  together  less  intimately ;  that 
is,  to  the  liquid  form.  In  other  words,  the  energy  which  existed 
in  the  flame  as  the  kinetic  energy  of  molecular  motion  has 
been  transformed,  upon  passage  into  the  melting  solid,  into 
the  potential  energy  of  molecules  which  have  been  pulled 
apart  against  the  force  of  their  mutual  attraction.  The  number 
of  calories  of  heat  energy  required  to  melt  one  gram  of  any  sub- 
stance without  producing  any  change  in  its  temperature  is  called 
the  heat  of  fusion  of  that  substance. 

*This  subject  should  be  preceded  by  a  laboratory  exercise  on  the  curve  of 
cooling  through  the  point  of  fusion,  and  followed  by  a  determination  of  the  heat 
of  fusion  of  ice.  See,  for  example,  Experiments  21  and  22  of  the  authors'  manual 


CHANGE  OF  STATE  — FUSION  171 

220.  Numerical  value  of  heat  of  fusion  of  ice.    Since  it  is 
found  to  require  about  80  times  as  long  for  a  given  flame  to 
melt  a  quantity  of  snow  as  to  raise  the  melted  snow  through 
1°  C.,  we  conclude  that  it  requires  about  80  calories  of  heat 
to  melt  1  g.  of  snow  or  ice.    This  constant  is,  however,  much 
more  accurately  determined  by  the  method  of  mixtures.   Thus, 
suppose  that  a  piece  of  ice  weighing,  for  example,  131  g.  is 
dropped  into  500  g.  of  water  at  40°  C.,  and  suppose  that  after 
the  ice  is  all  melted  the  temperature  of  the  mixture  is  found 
to  be  15°  C.    The  number  of  calories  which  have  come  out  of 
the  water  is  500  x  (40 -15)  =  12,500.    But  131x15=1965 
calories  of  this  heat  must  have  been  used  in  raising  the  ice 
from  0°  C.  to  15°  C.  after  it,  by  melting,  became  water  at  0°. 
The  remainder  of  the  heat,  namely  12,500-1965=10,535, 
must  have  been  used  in  melting  the  131  g.  of  ice.    Hence  the 
number  of  calories  required  to  melt  1  g.  of  ice  is  1  ^  g  ^  5  =  80.4. 

To  state  the  problem  algebraically,  let  x  =  the  heat  of  fusion 
of  ice.  Then  we  have 

131  x  + 1965  =12,500;  that  is,  x  =  80.4. 
According  to  the  most  careful  determination's  the  heat  of  fusion 
of  ice  is  80.0  calories. 

221.  Heat  given  out  when  water  freezes.  'Let  snow  and  salt  be 

added  to  a  beaker  of  water  until  the  temperature  of  the  liquid  mixture 
is  as  low  as  — 10°  C.  or  — 12°  C.  Then  let  a  test  tube  containing  a  ther- 
mometer and  a  quantity  of  pure  water  be  thrust  into  the  cold  solution. 
If  the  thermometer  is  kept  very  quiet,  the  temperature  of  the  water  in  the 
test  tube  will  fall  four  or  five,  or  even  t^n,  degrees  below  0°C.  without 
producing  solidification.  But  as  soon  as  the  thermometer  is  stirred,  or  a 
small  crystal  of  ice  is  dropped  into  the  neck  of  the  tube,  the  ice  crystals 
will  form  with  great  suddenness,  and  at  the  same  time  the  thermometer 
will  rise  to  0°  C.,  where  it  will  remain  until  all  the  water  is  frozen. 

The  experiment  shows  in  a  very  striking  way  that  the  proc- 
ess of  freezing  is  a  heat-evolving  process.  This  was  to  have 
been  expected  from  the  principle  of  the  conservation  of  energy,- 


172  WORK  AND  HEAT  EKEEGY 

for  since  it  takes  80  calories  of  heat  energy  to  turn  a  gram  of  ice 
at  0°  O.  into  water  at  0°  (7.,  this  amount  of  energy  must  reappear 
when  the  water  turns  back  to  ice. 

222.  Utilization  of  heat  evolved  in  freezing.    The  heat  given 
off  by  the  freezing  of  water  is  often    turned  to   practical 
account;  for  example,  tubs  of  water  are  sometimes  placed 
in  vegetable  cellars  to  prevent  the  vegetables  from  freezing. 
The  effectiveness  of  this  procedure  is  due  to  the  fact  that  the 
temperature  at  which  the  vegetables  freeze  is  slightly  lower 
than  0°  C.    As  the  temperature  of  the  cellar  falls  the  water 
therefore  begins  to  freeze  first,  and  in  so  doing  evolves  enough 
heat  to  prevent  the  temperature  of  the  room  from  falling  as 
far  below  0°  C.  as  it  otherwise  would. 

It  is  partly  because  of  the  heat  evolved  by  the  freezing  of 
large  bodies  of  water  that  the  temperature  never  falls  so  low 
in  the  vicinity  of  large  lakes  as  it  does  in  inland  localities. 

223.  Latent  heat.    Before  the  time  of  Joule,  when  heat  was 
supposed  to  be  a  weightless  fluid,  the  heat  which  disappears  in 
a  substance  when  it  melts  and  reappears  again  when  it  solidifies 
was  called  latent  or  hidden  heat.    Thus  water  was  said  to  have 
a  latent  heat  of  80  calories.    This  expression  is  still  in  common 
use,  although,  with  the  change  which  has  taken  place  in  our 
views  of  the  nature  of  heat,  its  appropriateness  is  entirely 
gone.    For  the  heat  energy  which  is  required  to  change  a  sub- 
stance from  a  solid  to  a  liquid  does  not  exist  within  the  liquid 
as  concealed  or  hidden  heat  energy,  but  has,  instead,  ceased  to 
exist  as  heat  energy  at  all^  having  been  transformed  into  the 
potential  energy  of  partially   separated  molecules ;   that  is, 
latent  heat  represents  the  work  which  has  been  done  in  effecting 
the  change  of  state. 

224.  Melting  points  of  crystalline  substances.    If  a  piece  of 
ice  is  placed  in  a  vessel  of  boiling  water  for  an  instant  and 
then  removed  and  wiped,  it  will  not  be  found  to  be  in  the 
slightest  degree  warmer  than  a  piece  of  ice  which  has  not  been 


CHANGE  OF  STATE  — FUSION  173 

exposed  to  the  heat  of  the  warm  water.  The  melting  point  of 
ice  is  therefore  a  perfectly  fixed,  definite  temperature,  above 
which  the  ice  can  never  be  raised  so  long  as  it  remains  ice,  no 
matter  how  fast  heat  is  applied  to  it.  All  crystalline  sub- 
stances are  found  to  behave  exactly  like  ice  in  this  respect, 
each  substance  of  this  class  having  its  characteristic  melting 
point.  The  following  table  gives  the  melting  points  of  some 
of  the  commoner  crystalline  substances : 


Mercury  . 
Ice  .  .  . 
Benzine  . 

-  39.5° 
0 

7 

C. 
it 

1C 

Sulphur 
Tin      .    .    . 
Lead       .    . 

.    114°  C. 

.    233  " 
.    330  " 

Silver 
Copper     . 
Cast  iron 

.      954°  C. 
.    1100  " 
.    1200  " 

Acetic  acid 

17 

" 

Zinc         .    . 

.    433  " 

Platinum 

.    1775  " 

Paraffin 

.    54 

" 

Aluminium 

.    650  " 

Iridium    . 

1950  " 

We  may  summarize  the  experiments  upon  melting  points  of 
crystalline  substances  in  the  two  following  laws : 

1.  The  temperatures  of  solidification  and  of  fusion  are  the  same. 

2.  The  temperature  of  the  melting  or  solidifying  substance 
remains  constant  from  the  moment  at  which  melting  or  solidi- 
fication begins  until  the  process  is  completed. 

\/ 

225.  Fusion  of  noncrystalline  or  amorphous  substances.  Let 
the  end  of  a  glass  rod  be  held  in  a  Bunsen  flame.  Instead  of  changing 
suddenly  from  the  solid  to  the  liquid  state,  it  will  gradually  grow  softer 
and  softer  until,  if  the  flame  is  sufficiently  hot,  a, drop  of  molten  glass 
will  finally  fall  from  the  end  of  the  rod. 

If  the  temperature  of  the  rod  had  been  measured  during 
this  process,  it  would  have  been  found  to  be  continually  rising. 
This  behavior,  so  completely  unlike  that  of  crystalline  sub- 
stances, is  characteristic  of  tar,  wax,  resin,  glue,  gutta-perchay 
alcohol,  carbon,  and  in  general  of  all  amorphous  substances. 
Such  substances  cannot  be  said  to  have  any  definite  melting 
points  at  all,  for  they  pass  through  all  stages  of  viscosity  both 
in  melting  and  in  solidifying.  It  is  in  virtue  of  this  property 
that  glass  and  other  similar  substances  can  be  heated  to  soft- 
ness and  then  molded  or  rolled  into  any  desired  shapes. 


1T4  WORK  AND  HEAT  ENERGY 

226.  Change  of  volume  on  solidifying.  One  has  only  to 
reflect  that  ice  floats,  or  that  bottles  or  crocks  of  water  burst 
when  they  freeze,  in  order  to  know  that  water  expands  upon 
solidifying.  In  fact,  1  cubic  foot  of  water  becomes  1.09  cubic 
feet  of  ice,  thus  expanding  more  than  one  twelfth  of  its  initial 
volume  when  it  freezes.  This  may  seem  strange  in  view  of 
the  fact  that  the  molecules  are  certainly  more  closely  knit 
together  in  the  solid  than  in  the  liquid  state ;  but  the  strange- 
ness disappears  when  we  reflect  that  in  freezing  the  molecules 
of  water  group  themselves  into  crystals,  and  that  this  operation 
presumably  leaves  comparatively  large  free  spaces  between 
different  crystals,  so  that,  although  groups  of  individual  mole- 
cules are  more  closely  joined  than  before,  the  total  volume 
occupied  by  the  whole  assemblage  of  molecules  is  greater. 

But  the  great  majority  of  crystalline  substances  are  unlike 
water  in  this  respect,  for,  with  the  exception  of  antimony  and 
bismuth,  they  all  contract  upon  solidifying  and  expand  on 
liquefying.  It  is  only  from  substances  which  expand,  or  which, 
like  cast  iron  change  in  volume  very  little  on  solidifying,  that 
sharp  castings  can  be  made.  For  it  is  clear  that  contracting 
substances  cannot  retain  the  shape  of  the  mold.  It  is  for  this 
reason  that  gold  and  silver  coins  must  be  stamped  rather  than 
cast.  Any  metal  from  which  type  is  to  be  cast  must  be  one 
which  expands  upon  solidifying,  for  it  need  scarcely  be  said 
that  perfectly  sharp  outlines  are  indispensable  to  good  type. 
Ordinary  type  metal  is  an  alloy  of  lead,  antimony,  and  copper 
which  fulfills  these  requirements. 

•  227.  Effect  of  the  expansion  which  water  undergoes  on 
freezing.  If  water  were  not  unlike  most  substances  in  that  it 
expands  on  freezing,  many,  if  not  all,  of  the  forms  of  life  whi^h 
now  exist  on  the  earth  would  be  impossible.  For  in  winter 
the  ice  would  sink  in  ponds  and  lakes  as  fast  as  it  froze,  and 
soon  our  rivers,  lakes,  and  perhaps  our  oceans  also  would 
become  solid  ice. 


CHANGE  OF  STATE  — FUSION  175 

The  force  exerted  by  the  expansion  of  freezing  water  is  very 
great.  Steel  bombs  have  been  burst  by  filling  them  with  water 
and  exposing  them  on  cold  winter  nights.  One  of  the  chief 
agents  in  the  disintegration  of  rocks  is  the  freezing  and  conse- 
quent expansion  of  water  which  has  percolated  into  them. 

228.  Pressure  lowers  the  melting  point  of  substances  which 
expand  on  solidifying.  Since  the  outside  pressure  acting  on 
the  surface  of  a  body  tends  to  prevent  its  expansion,  we  should 
expect  that  any  increase  in  the  outside  pressure  would  tend 
to  prevent  the  solidification  of  substances  which  expand  upon 
freezing.  It  ought  therefore  to  require  a  lower  temperature 
to  freeze  ice  under  a  pressure  of  two  atmospheres  than  under 
a  pressure  of  one.  Careful  experi- 
ments have  verified  this  conclusion, 
and  have  shown  that  the  melting 
point  of  ice  is  lowered  .0075°  C.  for 
an  increase  of  one  atmosphere  in 
the  outside  pressure.  Although  this 
lowering  is  so  small  a  quantity,  its 
existence  may  be  shown  as  follows : 

Let  two  pieces  of  ice  be  pressed  firmly  FIG.  179.    Regelation 

together  beneath  the  surface  of  a  vessel 

full  of  warm  water.  When  taken  out  they  will  Ibe  found  to  be  frozen 
together,  in  spite  of  the  fact  that  they  have  been  immersed  in  a  medium 
much  warmer  than  the  freezing  point  of  water.  The  explanation  is 
as  follows : 

At  the  points  of  contact  the  pressure  reduces  the  freezing  point  of 
the  ice  below  0°C.,  and  hence  it  melts  and  gives  rise  to  a  thin  film  of 
water  the  temperature  of  which  is  slightly  below  0°C.  When  this 
pressure  is  released  the  film  of  water  at  once  freezes,  for  its  tempera- 
ture is  below  the  freezing  point  corresponding  to  ordinary  atmospheric 
pressure.  The  same  phenomenon  may  be  even  more  strikingly  illus- 
trated by  the  following  experiment : 

Let  two  weights  of  from  5  to  10  kg.  be  hung  by  a  wire  over  a  block 
of  ice  as  in  Fig.  179.  In  half  an  hour  or  less  the  wire  will  be  found  to 
have  cut  completely  through  the  block,  leaving  the  ice,  however,  as 


176  WORK  AND  HEAT  ENERGY 

solid  as  at  first.  The  explanation  is  as  follows:  Just  below  the  wire 
the  ice  melts  because  of  the  pressure ;  as  the  wire  sinks  through  the 
layer  of  water  thus  formed,  the  pressure  on  the  water  is  relieved  and  it 
immediately  freezes  again  above  the  wire. 

This  process  of  melting  under  pressure  and  freezing  again 
as  soon  as  the  pressure  is  relieved  is  known  as  revelation. 

229.  Pressure  raises  the  freezing  point  of  substances  which 
contract  on  solidifying.  Substances  like  paraffin,  zinc,  and  lead 
which  contract  on  solidifying  have  their  melting  points  raised 
by  an  increase  in  pressure,  for  in  this  case  the  outside  pressure 
tends  to  assist  the  molecular  forces  which  are  pulling  the  body 
out  of  the  larger  liquid  form  into  the  smaller  solid  form ;  hence 
this  result  can  be  accomplished  at  a  higher  temperature  with 
pressure  than  without  it. 

We  may  therefore  summarize  the  effects  of  pressure  on  the 
melting  points  of  crystalline  substances  in  the  following  law : 

Substances  which  expand  on  solidifying  have  their  melting 
points  lowered  by  pressure,  and  those  which  contract  on  solidify- 
ing have  their  melting  points  raised  by  pressure. 

QUESTIONS  AND  PROBLEMS 

1.  What  is  the  temperature  of  a  mixture  of  ice  and  water?    What 
determines  whether  it  is  freezing-  or  melting  ? 

2.  Why  does  ice  cream  seem  so  much  colder  to  the  teeth  than  ice- 
water  ? 

3.  What  is  the  meaning  of  the  statement  that  the  heat  of  fusion  of 
mercury  is  2.8? 

4.  Why  will  a  snowball  "pack  "  if  the  snow  is  melting,  but  not  if  it 
is  much  below  0°  C.  ? 

5.  If   water  were  like  gold  in  contracting  on  solidification,  what 
would  happen  to  lakes  and  rivers  during  a  cold  winter  ? 

6.  Equal  weights  of  hot  water  and  ice  are  mixed  and  the  result  is 
water  at  0°C.    What  was  the  temperature  of  the  hot  water? 

7.  Which  is  the  more  effective  as  a  cooling  agent,  100  Ib.  of  ice  at 
0°C.  or  100  Ib.  of  water  at  the  same  temperature?  Why? 

8.  How  many  grams  of  ice  must  be  put  into  500  g.  of  water  at  50°  C. 
t©  lower  the  temperature  to  10° C.? 


CHANGE  OF  STATE  —  VAPORIZATION  177 

CHANGE  OF  STATE  —  VAPORIZATION  * 

230.  Heat  of  vaporization  defined.    The  experiments  per- 
formed in  Chapter  IV,  on  Molecular  Motions,  led  us  to  the 
conclusion  that,  at  the  free  surface  of  any  liquid,  molecules 
frequently  acquire  velocities  sufficiently  high  to  enable  them 
to  lift  themselves  beyond  the  range  of  attraction  of  the  mole- 
cules of  the  liquid  and  to  pass  off  as  free  gaseous  molecules 
into  the  space  above.    They  taught  us  further  that  since  it  is 
only  suchnnolecules  as  have  unusually  high  velocities  which 
are  able  thus  to  escape,  the  average  kinetic  energy  of  the  mole- 
cules left  behind  is  continually  diminished  by  this  loss  from 
the  liquid  of  the  most  rapidly  moving  molecules,  and  conse- 
quently the  temperature  of  an  evaporating  liquid  constantly 
falls  until  the  rate  at  which  it  is  losing  heat  is  equal  to  the 
rate  at  which  it  receives  heat  from  outside  sources.    Evapora- 
tion, therefore,  always  takes  place  at  the  expense  of  the  heat 
energy  of  the  liquid.    The  number  of  calories  of  heat  which  dis- 
appear in  the  formation  of  one  gram  of  vapor  is  called  the  heat 
of  vaporization  of  the  liquid. 

231.  Heat  due  to  condensation.    When  molecules  pass  off 
from  the  surface  of  a  liquid  they  rise  against  the  downward 
forces  exerted  upon  them  by  the  liquid?  and  in  so  doing  ex- 
change a  part  of  their  kinetic  energy  for  the  potential  energy  of 
separated  molecules  in  precisely  the  same  way  in  which  a  ball 
thrown  upward  from  the  earth  exchanges  its  kinetic  energy 
in  rising  for  the  potential  energy  which  is  represented  by  the 
separation  of  the  ball  from  the  earth.    Similarly,  just  as  when 
the  ball  falls  back,  it  regains  in  the  descent  all  of  the  kinetic 
energy  lost  in  the  ascent,  so  when  the  molecules  of  the  vapor 

*  It  is  recommended  that  this  subject  be  accompanied  by  a  laboratory  deter- 
mination of  the  boiling  point  of  alcohol  by  the  direct  method,  and  by  the  vapor- 
pressure  method,  and  that  it  be  followed  by  an  experiment  upon  the  fixed  points 
of  a  thermometer  and  the  change  of  boiling  point  with  pressure.  See,  for  example, 
Experiments  23  and  24  of  the  authors'  manual. 


178 


WOEK  AND  HEAT  EKEEGY 


reenter  the  liquid,  they  must  regain  all  of  the  kinetic  energy 
which  they  lost  when  they  passed  out  of  the  liquid.  We  may 
expect,  therefore,  that  every  gram  of  steam  which  condenses  will 
generate  in  this  process  the  same  number  of  calories  which  was 
required  to  vaporize  it. 

232.  Measurement  of  heat  of  vaporization.   To  find  accurately 

the  number  of  calories  expended  in  the  vaporization,  or  released  in  the 
condensation,  of  a  gram  of  water  at  100°  C.,  we  pass  steam  rapidly 
for  two  or  three  minutes  from  an  arrangement  like  that  shown  in 
Fig.  180  into  a  vessel  containing,  say, 
500  g.  of  water.  We  observe  the  initial  and 
final  temperatures  and  the  initial  and  final 
weights  of  the  water.  If,  for  example,  the 
gain  in  weight  of  the  water  is  16.5  g.,  we 
know  that  16.5  g.  of  steam  have  been  con- 
densed. If  the  rise  in  temperature  of  the 
water  is  from  10°  C.  to  30°  C.,  we  know  that 
500  x  (30  -  10)  =  10,000  calories  of  heat 
have  entered  the  water.  If  x  represents 
the  number  of  calories  given  up  by  1  g.  of 
steam  in  condensing,  then  the  total  heat 
imparted  to  the  water  by  the  condensation 
of  the  steam  is  16.5  x  calories.  This  condensed  steam  is  at  first  water  at 
100°  C.,  which  is  then  cooled  to  30°  C.  In  this  cooling  process  it  gives  up 
16.5  X  (100  —  30)  =  1155  calories.  Therefore,  equating  the  heat  gained 
by  the  water  to  the  heat  lost  by  the  steam,  we  have 

10,000  =  16.5  x  +  1155,  or  x  =  536.1. 

This  is  the  method  usually  employed  for  finding  the  heat  of 
vaporization.    The  now  accepted  value  of  this  constant  is  536. 

233.  Boiling  temperature  defined.    If  a  liquid  is  heated  by 
means  of  a  flame,  it  will  be  found  that  there  is  a  certain  tem- 
perature above  which  it  cannot  be  raised,  no  matter  how  rapidly 
the  heat  is  applied.    This  is  the  temperature  which  exists  when 
bubbles  of  vapor  form  at  the  bottom  of  the  vessel  and  rise  to 
the  surface,  growing  in  size  as  they  rise.    This  temperature  is 
commonly  called  the  boiling  temperature. 


FIG.  180.    Heat  of  vaporiza- 
tion of  water 


CHANGE  OF  STATE  —  VAPORIZATION 

But  a  second  and  more  exact  definition  of  the  boiling  point 
may  be  given.  It  is  clear  that  a  bubble  of  vapor  can  exist 
within  the  liquid  only  when  the  pressure  exerted  by  the  vapor 
within  the  bubble  is  at  least  equal  to  the  atmospheric  pressure 
pushing  down  on  the  surface  of  the  liquid.  For  if  the  pres- 
sure within  the  bubble  were  less  than  the  outside  pressure, 
the  bubble  would  immediately  coljapse.  Therefore,  the  boiling 
point  is  the  temperature  at  which  the  pressure  of  the  saturated 
vapor  first  becomes  equal  to  the  pressure  existing  outside. 

234.  Variation  of  the  boiling  point  with  pressure.  Since 
the  pressure  of  a  saturated  vapor  varies  rapidly  with  the 
temperature,  and  since  the  boiling  point  has  been  denned 
as  the  temperature  at  which  the  pres- 
sure of  the  saturated  vapor  is  equal  to 
the  outside  pressure,  it  follows  that  the 
boiling  point  must  vary  as  the  outside  pres- 
sure varies. 


Thus    let   a   round-bottomed  flask  be    half 
filled  with  water  and  boiled.    After  the  boiling 
has  continued  for  a  few  minutes,  so  that  the 
steam  has  driven  out  most  of  the  air  from  the     -pIG  ^g^    Lowering  the 
flask,  let  a   rubber    stopper  be    inserted,   and     boiling  point  by  dimin- 
the  flask  removed  from  the  flame  and  inverted         ishing  the  pressure 
as  shown  in  Fig.  181.    The  temperature  will 

fall  rapidly  below  the  boiling  point.  But  if  cold  water  is  poured  over  the 
flask,  the  water  will  again  begin  to  boil  vigorously,  for  the  cold  water, 
by  condensing  the  steam,  lowers  the  pressure  within  the  flask,  and  thus 
enables  the  water  to  boil  at  a  temperature  lower  than  100°  C.  The  boil- 
ing will  cease,  however,  as  soon  as  enough  vapor  is  formed  to  restore  the 
pressure.  The  operation  may  be  repeated  many  times  without  reheating. 

At  the  city  of  Quito,  Ecuador,  water  boils  at  90°  C.,  and 
on  the  top  of  Mt.  Blanc  it  boils  at  84°  C.  On  the  other 
hand,  in  the  boiler  of  a  steam  engine  in  which  the  pressure 
is  100  Ib.  to  the  square  inch,  the  boiling  point  of  the  water 
is  155°  C. 


180  WORK  AND  HEAT  ENERGY 

235.  Evaporation  and  boiling.    The  only  essential  difference 
between  evaporation  and  boiling  is  that  the  former  consists  in 
the  passage  of  molecules  into  the  vaporous  condition  from  the 
free  surface  only,  while  the  latter  consists  in  the  passage  of  the 
molecules  into  the  vaporous  condition  both  at  the  free  surface 
and  at  the  surface  of  bubbles  which  exist  within  the  body  of 
the  liquid.    The  only  reason,  that  vaporization  takes  place  so 
much  more  rapidly  at  the  boiling  temperature  than  just  below 
it  is  that  the  evaporating  surface  is  enormously  increased  as 
soon  as  the  bubbles  form.    The  reason  the  temperature  cannot 
be  raised  above  the  boiling  point  is  that  the  surface  always 
increases,  on  account  of  the  bubbles,  to  just  such  an  extent 
that  the  loss  of  heat  because  of  evaporation  is  exactly  equal 
to  the  heat  received  from  the  fire. 

236.  Distillation.    Let  water  holding  in  solution  some  aniline  dye 
be  boiled  in  B  (Fig.  182).    The  vapor  of  the  liquid  will  pass  into  the 
tube   T,  where  it  will  be  condensed  by  the  cold  water  which  is  kept 
in  continual  circulation  through  the  jacket  J.    The  condensed  water 
collected  in  P  will  be  seen  to  be  free 

from   all  traces  of  the  color  of  the 
dissolved  aniline. 

We  learn  then  that  tvhen  solids 
are  dissolved  in  liquids  the  vapor 
which  rises  from  the  solution  con- 
tains none  of  the  dissolved  sub- 
stance. Sometimes  it  is  the  pure 
liquid  in  P  which  is  desired,  as  "  FJG  182  Distillation 
in  the  manufacture  of  alcohol, 

and  sometimes  the  solid  which  remains  in  B,  as  in  the  manu- 
facture of  sugar.  In  the  white-sugar  industry  it  is  necessary 
that  the  evaporation  take  place  at  a  low  temperature,  so  that 
the  sugar  may  not  be  scorched.  Hence  the  boiler  is  kept  par- 
tially exhausted  by  means  of  an  air  pump,  thus  enabling  the 
solution  to  boil  at  considerably  reduced  temperatures. 


CHANGE  OF  STATE  —  VAPORIZATION          181 

237.  Fractional  distillation.  When  both  of  the  constituents 
of  a  solution  are  volatile,  as  in  the  case  of  a  mixture  of  alcohol 
and  water,  the  vapor  of  both  will  rise  from  the  liquid.  But 
the  one  which  has  the  lower  boiling  point,  that  is,  the  higher 
vapor  pressure,  will  predominate.  Hence,  if  we  have  in  B 
(Fig.  182)  a  solution  consisting  of  50%  alcohol  and  50%  water, 
it  is  clear  that  we  can  obtain  in  P,  by  evaporating  and  con- 
densing, a  solution  containing  a  much  larger  percentage  of 
alcohol.  By  repeating  this  operation  a  number  of  times  we 
can  increase  the  purity  of  the  alcohol.  This  process  is  called 
fractional  distillation.  The  boiling  point  of  the  mixture  lies 
between  the  boiling  points  of  alcohol  and  water,  being  higher 
the  greater  the  percentage  of  water  in  the  solution. 

QUESTIONS  AND  PROBLEMS 

1.  After  water  has  been  "  brought  to  a  boil,"  will  eggs  become  hard 
any  quicker  when  the  flame  is  high  than  when  it  is  low  V 

2.  The  hot  water  which  leaves  a  steam  radiator  may  be  as  hot  as 
the  steam  which  entered  it.    How  then  has  the  room  been  warmed  ? 

3.  How  much  heat  is  given  up  by  30  g.  of  steam  at  100°  C.  in  con- 
densing to  water  at  the  same  temperature  ? 

4.  A  vessel  contains  300  g.  of  water  at  0°C.  and  130  g.  of  ice. 
If    25  g.   of   steam   are  condensed  in  it,   what  will    be    the  resulting 
temperature  ? 

5.  How  many  calories  are  given  up  by  30  g.  of  steam  at  100°  C.  in 
condensing  and  then  cooling  to  20°  C.  ?    How  much  water  will  this  steam 
raise  from  10° C.  to  20°  C.? 

6.  Why  do  fine  bubbles  rise  in  a  vessel  of  water  which  is  being 
heated  long  before  the  boiling  point  is  reached  ?  How  can  you  distinguish 
between  this  phenomenon  and  boiling  ? 

7.  When  water  is  boiled  in  a  deep  vessel  it  will  be  noticed  that  the 
bubbles  rapjdly  increase  in  size  as  they  approach  the  surface.    Give  two 
reasons  for  this. 

8.  Why  does  steam  produce  so  much  more  severe  burns  than  hot 
water  of  the  same  temperature  ? 

9.  Why  in  winter  does  not  all  the  snow  melt  at  once  as  soon  as  the 
temperature  of  the  air  rises  above  0°  C.? 

10.  Explain  how  freezing  and  thawing  disintegrate  rocks. 


182  WORK  AND  HEAT  ENERGY 

11.  In  the  fall  we  expect  frost  on  clear  nights  when  the  dew  point  is 
low,  but  not  on  cloudy  nights  when  the  dew  point  is  high.    Can  you  see 
any  reason  why  a  large  deposit  of  dew  will  prevent  the  temperature  of 
the  air  from  falling  very  low? 

12.  Why  does  the  (Distillation  of  a  mixture  of  alcohol  and  water 
always  result  to  some  extent  'in  a  mixture  of*  alcohol  and  water  ? 

13.  A  fall  of  1°  C.  in  the  boiling  point  is  caused  by  rising  960  ft. 
How  hot  is  boiling  water  at  Denver,  5000  ft.  above  sea  level  ? 

14.  How  may  we  obtain  pure  drinking  water  from  sea  water? 

^KTJIFICIAL  COOLING 

238.  Cooling  by  solution.  Let  a  handful  of  common  salt  be  placed 
in  a  small  beaker  of  water  at  the  temperature  of  the  room  and  stirred 
with  a  thermometer.  The  temperature  will  fall  several  degrees.  If  equal 
weights  of  ammonium  nitrat^  and  water  at  15°  C.  are  mixed,  the  temper- 
ature will  fall  as  low  as  —  10°  C.  If  the  water  is  nearly  at  0°  C.  when  the 
ammonium  nitrate  is  added,  and  if  the  stirring  is  done  with  a  test  tube 
partly  filled  with  ice-cold  water,  the  water  in  the  tube  will  be  frozen. 

These  experiments  show  that  the  breaking  up  of  the  crystals 
of  a  solid  requires  an  expenditure  of  heat  energy,  as  well  when 
this  operation  is  effected  by  solution  as  by  fusion.  The  reason 
for  this  will  appear  at  once  if  we  consider  the  analogy  between 
solution  and  evaporation.  For  just  as  the  molecules  of  a  liquid 
tend  to  escape  from  its  surface  into  the  air,  so  the  molecules  at 
^he  ^urface  of  the  salt  are  tending,  because  of  their  velocities, 
'to  pass  off,  and  are  only  held  back  by  the  attractions  of  the 
other  molecules  in  the  crystal  to  which  they  belong.  If,  how- 
ever, the  salt  is  placed  in  water,  the  attraction  of  the  water 
molecules  for  the  salt  molecules  aids  the  natural  velocities  of 
the  latter  to  carry  them  beyond  the  attraction  of  their  fellows. 
As  the  molecules  escape  from  the  salt  crystals  two  forces  are 
acting  on  them,  the  attraction  of  the  water  molecules  tending 
to  increase  their  velocities,  and  the  attraction  of  the  remaining 
salt  molecules  tending  to  diminish  these  velocities.  If  the 
latter  force  has  a  greater  resultant  effect  than  the  former,  the 
mean  velocity  of  the  molecules  after  they  have  escaped  will 


ARTIFICIAL  COOLING  183 

be  diminished  and  the  solution  will  be  cooled.  But  if  the 
attraction  of  the  water  molecules  amounts  to  more  than  the 
backward  pull  of  the  dissolving  molecules,  as  when  caustic 
potash  or  sulphuric  acid  is  dissolved,  the  mean  molecular 
velocity  is  increased  and  the  solution  is  heated. 

239.  Freezing  points  of  solutions.    If  a  solution  of  one  part 
of  common  salt  to  ten  of  water  is  placed  in  a  test  tube  and 
immersed  in  a  "  freezing  mixture  "  of  water,  ice,  and  salt,  the 
temperature  indicated  by  a  thermometer  in  the  tube  will  not  be 
zero  when  ice  begins  to  form,  but  several  degrees  below  zero. 
The  ice  which  does  form,  however,  will  be  found,  like  the  vapor 
which  rises  above  a  salt  solution,  to  be  free  from  salt,  and  it  is 
this  fact  which  furnishes  a  key  to  the  explanation  of  why  the 
freezing  point  of  the  salt  solution  is  lower  than  that  of  pure 
water.    For  cooling  a  substance,  to  its  freezing  point  simply 
means  reducing  its  temperature,  and  therefore  the  mean  ve- 
locity of  its  molecules,  sufficiently  to  enable  the  cohesive  forces 
of  the  liquid  to  pull  the  molecules  together  into  the  crystalline 
form.    Since  in  the  freezing  of  a  salt  solution  the  cohesive 
forces  of  the  water  are  obliged  to  overcome  the  attractions 
of  the  salt  molecules  as  well  as  the  molecular  motions,  the 
motions  must  be  rendered  less,  that  is,  the  temperature  must 
be  made  lower,  than  in  the  case  of  pure  water  in  order  that 
crystallization  may  occur.    We  should  expect  from  this  reason- 
ing that  the  larger  the  amount  of  salt  in  solution  the  lower 
would  be  the  freezing  point.    This  is  indeed  the  case.    The 
lowest  freezing  point  obtainable  with  common  salt  in  water  is 
-  22°  C.    This  is  the  freezing  point  of  a  saturated  solution. 

240.  Freezing  mixtures.    If  snow  or  ice  is  placed  in  a  vessel 
of  water,  the  water  melts  it,  and  in  so  doing  its  temperature  is 
reduced  to  the  freezing  point  of  pure  water.    Similarly,  if  ice 
is  placed  in  salt  water,  it  melts  and  reduces  the  temperature  of 
the  salt  water  to  the  freezing  point  of  the  solution.   This  may 
be  one,  or  two,  or  twenty-two  degrees  below  zero,  according 


184 


WOEK  AND  HEAT  ENEEGY 


to  the  concentration  of  the  solution.  Whether  then  we  put 
the  ice  in  pure  water  or  in  salt  water,  enough  of  it  always 
melts  to  reduce  the  whole  mass  to  the  freezing  point  of  the 
liquid,  and  each  gram  of  ice  which  melts  uses  up  80  calories 
of  heat.  The  efficiency  of  a  mixture  of  salt  and  ice  in  producing 
cold  is  therefore  due  simply  to  the  fact  that  the  freezing  point  of 
a  salt  solution  is  lower  than  that  of  pure  water. 

The  best  proportions  are  three  parts  of  snow  or  finely  shaved 
ice  to  one  part  of  common  salt.  If  three  parts  of  calcium 
chloride  are  mixed  with  two  parts  of  snow,  a  temperature 
of  —  55°  C.  may  be  produced.  This  is 
low  enough  to  freeze  mercury. 

241.  Intense  cold  by  evaporation.  If, 
instead  of  utilizing  as  above  the  heats 
of  fusion,  we  utilize  the  larger  heats  of 
vaporization,  still  lower  temperatures 
may  be  produced  (see  §§92  and  161). 

Thus,  if  a  cylinder  of  liquid  carbon  diox- 
ide is  placed  as  in  Fig.  183  and  the  stopcock 
opened,  such  intense  cold  is  produced  by  the 
rapid  evaporation  .of  the  liquid  which  rushes 
out  into  the  bag  that  the  liquid  freezes  to  a 
snowy  solid.  The  solid  itself  evaporates  so 
rapidly  that  it  maintains,  as  long  as  it  lasts, 

a  temperature  of  —  80°  C.  If  a  little  of  this  solid  is  placed  in  a  beaker 
containing  ether,  and  the  mixture  is  stirred  with  a  test  tube  filled  with 
mercury,  the  mercury  will  be  frozen  solid.  The  chief  function  of  the  ether 
is  to  insure  intimate  contact  between  the  cold  solid  and  the  test  tube. 

QUESTIONS  AND  PROBLEMS 

1.  Explain  why  salt  is  sometimes  thrown  on  icy  sidewalks  on  cold 
winter  days. 

2.  When  salt  water  freezes  the  ice  formed  is  practically  free  from 
salt.    What  effect,  then,  does  freezing  have  on  the  concentration  of  a 
salt  solution? 

3.  A  partially  concentrated  salt  solution  which  has  a  freezing  point  of 
—  5°  C.  is  placed  in  a  room  which  is  kept  at  —  10°  C.    Will  it  all  freeze? 


FIG.  183.  Cold  by  rapid 

evaporation  of  carbon 

dioxide 


INDUSTRIAL  APPLICATIONS  185 

4.  Give  two  reasons  why  the  ocean  freezes  less  easily  than  the  lakes. 

5.  Why  does  pouring  H2SO4  into  water  produce  heat,  while  pouring 
the  same  substance  upon  ice  produces  cold? 

6.  It  sometimes  happens  that  a  liquid  which  is  unable  to  dissolve  a 
solid  at  a  low  temperature  will  do  so  at  a  higher  temperature.    Why? 
(See  §  238.) 

7.  When  the  salt  in  an  ice-cream  freezer  unites  with  the  ice  to  form 
brine,  about  how  many  calories  of  heat  are  used  for  each  gram  of  ice 
melted  ?    Where  does  it  come  from  ?    If  the  freezing  point  of  the  salt 
solution  were  the  same  as  that  of  the  cream,  would  the  latter  freeze  ? 


INDUSTRIAL  APPLICATIONS 

242.  The  modern  steam  engine.  Thus  far  in  our  study  of 
the  transformations  of  energy  we  have  considered  only  cases 
in  which  mechanical  energy  was  transformed  into  heat  energy. 
In  all  heat  engines  we  have  examples  of  exactly  the  reverse 
operation,  namely,  the  transformation  of  heat  energy  back  into 
mechanical  energy.  How  this  is  done  may  best  be  understood 
from  a  study  of  various  modern  forms  of  heat  engines.  The 
invention  of  the  form  of  the  steam  engine  which  is  now  in  use 
is  due  to  James  Watt,  who,  at  the  time  of  the  invention  (1768), 
was  an  instrument  maker  in  the  University  of  Glasgow. 

The  operation  of  such  a  machine  can  best  be  understood  from 
the  ideal  diagram  shown  in  Fig.  184.  Steam  generated  in  the 
boiler  B  by  the  fire  F  passes  through  the  pipe  S  into  the  steam 
chest  V,  and  thence  through  the  passage  N  into  the  cylinder  (7, 
where  its  pressure  forces  the  piston  P  to  the  left.  It  will  be 
seen  from  the  figure  that,  as  the  driving  rod  It  moves  toward 
the  left,  the  so-called  eccentric  rod  R',  which  controls  the  valve  V, 
moves  toward  the  right.  Hence,  when  the  piston  has  reached 
the  left  end  of  its  stroke  the  passage  JVwill  have  been  closed, 
while  the  passage  M  will  have  been  opened,  thus  throwing  the 
pressure  from  the  right  to  the  left  side  of  the  piston,  and  at  the 
same  time  putting  the  right  end  of  the  cylinder,  which  is  full 
of  spent  steam,  into  connection  with  the  exhaust  pipe  E.  This 


186 


WORK  AND  HEAT  ENERGY 


operation  goes  on  continually,  the  rod  R'  opening  and  closing 
the  passages  M  and  JVat  just  the  proper  moments  to  keep  the 
piston  moving  back  and  forth  throughout  the  length  of  the 
cylinder.  The  shaft  carries  a  heavy  flywheel  W,  the  great 


FIG.  184.    Ideal  diagram  of  a  steam  engine 

inertia  of  which  insures  constancy  in  speed.  The  motion  of 
the  shaft  is  communicated  to  any  desired  machinery  by  means 
of  a  belt  which  passes  over  the  pulley  W. 

243.  Condensing  and  noncondensing  engines.  In  most  sta- 
tionary engines  the  exhaust  E  leads  to  a  condenser  which  con- 
sists of  a  chamber  Q,  into  which  plays  a  jet  of  cold  water  T, 
and  in  which  a  partial  vacuum  is  maintained  by  means  of  an 
air  pump.  In  the  best  engines  the  pressure  within  Q  is  not 
more  than  from  3  to  5  centimeters  of  mercury ;  that  is,  not  more 
than  a  pound  to  the  square  inch.  Hence  the  condenser  reduces 
the  back  pressure  against  that  end  of  the  piston  which  is  open 
to  the  atmosphere  from  15  pounds  down  to  1  pound,  and  thus 


INDUSTEIAL  APPLICATIONS  187 

increases  the  effective  pressure  which  the  steam  on  the  other 
side  of  the  piston  can  exert.  Since,  however,  the  addition  of 
the  condenser  makes  the  engine  more  expensive,  more  heavy, 
and  more  complicated,  it  is  generally  omitted  on  locomotives, 
and  on  other  engines  in  which  simplicity,  compactness,  and 
stability  are  of  more  importance  than  economy  of  fuel.  It  is 
obvious  that  if  a  noncondensing  engine  is  to  have  the  same 
effective  pressure  on  the  piston  head  as  a  condensing  engine, 
the  pressure  maintained  within  the  boiler  must  be  about 
15  pounds  higher.  For  this  reason  noncondensing  engines 
are  often  called  high-pressure  engines.  Such  engines  can  easily 
be  recognized  by  the  puffs  of  exhaust  steam  which  they  send 
out  into  the  atmosphere  at  each  stroke  of  the  piston. 

244.  The  eccentric.    In  practice  the  valve  rod  R'  is  not  attached 
as  in   the   ideal  engine   indicated   in  Fig.  184,  but   motion   is   com- 
municated to  it  by  a  so-called  eccentric.    This  consists  of  a  circular 
disk  K  (Fig.  185)  rigidly  attached  to  the  axle,  but  so  set  that  its 
center   does  not  coin- 
cide with  the  center  of 

the  axle  A.  The  disk  K 
rotates  inside  the  col- 
lar C  and  thus  commu- 
nicates to  the  eccentric 
rod  R'  a  back-and-forth 
motion  which  operates 
the  valve  V  in  such  a 
way  as  to  admit  steam 
through  M  and  N  at  FIG.  185.  The  eccentric 

the  proper  time. 

245.  The  boiler.    When  an  engine  is  at  work  steam  is  being  removed 
very  rapidly  from  the  boiler ;  for  example,  a  railway  locomotive  consumes 
from  3  to  6  tons  of  water  per  hour.    It  is  therefore  necessary  to  have 
the  fire  in  contact  with  as  large  a  surface  as  possible.    In  the  tubular 
boiler  this  end  is  accomplished  by  causing  the  flames  to  pass  through 
a  large  number  of  metal  tubes  immersed  in  water.    The  arrangement 
of  the  furnace  and  the  boiler  may  be  seen  from  the  diagram  of  a 
locomotive  shown  in  Fig.  186. 


188 


WORK  AND  HEAT  ENERGY 


246.  The  draft.  In  order  to  suck  the  flames  through  the  tubes  B  of 
the  boiler  a  powerful  draft  is  required.  In  locomotives  this  is  obtained 
by  running  the  exhaust  steam  from  the  cylinder  C  (Fig.  186)  into  the 
smokestack  E  through  the  blower  F.  The  strong  current  through  F 


FIG.  186.   Diagram  of  locomotive 

draws  with  it  a  portion  of  the  air  from  the  smoke  box  Z),  thus  producing 
within  D  a  partial  vacuum  into  which  a  powerful  draft  rushes  from  the 
furnace  through  the  tubes  B.  The  coal  consumption  of  an  ordinary 
locomotive  is  from  one-fourth  ton  to  <pne  ton  per  hour. 

In  stationary  engines  a  draft  is  obtained  by  making  the  smoke- 
stack very  high.  Since  in  this  case  the  pressure  which  is  forcing  the 
air  through  the  furnace  is  equal  to  the  difference 
in  the  weights  of  columns  of  air  of  unit  cross  sec- 
tion inside  and  outside  the  chimney,  it  is  evident 
that  this  pressure  will  be  greater  the  greater  the 
height  of  the  smokestack.  This  is  the  reason  for 
the  immense  heights  given  to  chimneys  in  large 
power  plants. 

247.  The  governor.   Fig.  187  shows  an  ingenious 
device  of  Watt's,  called  a  governor,  for  regulating 
automatically  the  speed  with  which  a  stationary 
engine  runs.    If  it  runs  too  fast,  the  heavy  rotat- 
ing balls  B  move  apart  and  upward,  and  in  so  doing  operate  a  valve 
which  partially  shuts  off  the  supply  of  steam  from  the  cylinder. 

248.  Compound  engines.    In  an  engine  which  has  but  a  single  cylin- 
der the  full  force  of  the  steam  has  not  been  spent  when  the  cylinder 
is  opened  to  the  exhaust.    The  waste  of  energy  which  this  entails  is 


FIG.  187.   The 
governor 


INDUSTRIAL  APPLICATIONS 


189 


FIG.  188.    Compound  engine  cylinders 


obviated  in  the  compound  engine  by  allowing  the  partially  spent  steam 
to  pass  into  a  second  cylinder  of  larger  area  than  the  first.  The  most 
efficient  of  modern  engines  have  three  and  sometimes  four  cylinders 
of  this  sort,  and  the  engines  are  accordingly  called  triple  or  quadruple 
expansion  engines.  Fig.  188  shows 
the  relation  between  any  two  suc- 
cessive cylinders  of  a  compound 
engine.  By  automatic  devices 
not  differing  in  principle  from 
the  eccentric,  valves  Cl,  D2,  and 
E2  open  simultaneously  and  thus 
permit  steam  from  the  boiler  to 
enter  the  small  cylinder  A,  while 
the  partially  spent  steam  in  the 
other  end  of  the  same  cylinder 
passes  through  D2  into  B,  and 
the  more  fully  exhausted  steam  in  the  upper  end  of  B  passes  out 
through  E2.  At  the  upper  end  of  the  stroke  of  the  pistons  P  and  P', 
C\  D2,  and  E2  automatically  close,  while  C2,  D\  and  El  simultaneously 
open  and  thus  reverse  the  direction  of  motion  of  both  pistons.  These 
pistons  are  attached  to  the  same  shaft. 

249.  Efficiency  of  a  steam  engine.  We  have  seen  that  it  is 
possible  to  transform  completely  a  given  amount  of  mechani- 
cal energy  into  heat  energy.  This  is  done  whenever  a  moving 
body  is  brought  to  rest  by  means  .of  a  frictional  resistance. 
But  the  inverse  operation,  namely,  that  of 'transforming  heat 
energy  into  mechanical  energy,  differs  in  this  respect,  that  it 
is  only  a  comparatively  small  fraction  of  the  heat  developed 
by  combustion  which  can  be  transformed  into  work.  For  it  is 
not  difficult  to  see  that  in  every  steam  engine  at  least  a  part 
of  the  heat  must  of  necessity  pass  over  with  the  exhaust  steam 
into  the  condenser  or  out  into  the  atmosphere.  This  loss  is  so 
great  that  even  in  an  ideal  engine  not  more  than  about  23% 
of  the  heat  of  combustion  could  be  transformed  into  work.  In 
practice  the  very  best  condensing  engines  of  the  quadruple- 
expansion  type  transform  into  mechanical  work  not  more  than 
17%  of  the  heat  of  combustion.  Ordinary  locomotives  utilize 


190 


WORK  AND  HEAT  ENERGY 


at  most  not  more  than  8%.  The  efficiency  of  a  heat  engine  is 
defined  as  the  ratio  between  the  heat  utilized,  or  transformed  into 
work,  and  the  total  heat  expended.  The  efficiency  of  the  best 
steam  engines  is  therefore  about  |J-,  or  75%,  of  that  of  an  ideal 
heat  engine,  while  that  of  the  ordinary  locomotive  is  only  about 
^-,  or  26%,  of  the  ideal  limit. 

250.  The  principle  of  the  gas  engine.  Within  the  last 
decade  gas  engines  have  begun  to  replace  steam  engines  to  a 
very  great  extent,  especially  for  small-power  purposes.  These 
engines  are  driven  by  properly  timed  explosions  of  a  mixture 
of  gas  and  air  occurring  within 
the  cylinder. 

Fig.  189  is  a  diagram  illus- 
trating the  four  stages  into 
which  it  is  convenient  to  divide 
the  complete  cycle  of  oper- 
ations which  goes  on  within 
such  an  engine.  Suppose  that 
the  heavy  flywheels  W  have 
already  been  set  in  motion.  As 
the  piston  p  moves  to  the  right 
in  the  first  stroke  (see  7)  the 
valve  E  opens  and  an  explosive 
mixture  of  gas  and  air  is  drawn 
into  the  cylinder  through  E. 
As  the  piston  returns  to  the  left  (see  £)  valve  E  closes,  and 
the  mixture  of  gas  and  air  is  compressed  into  a  small  space 
in  the  left  end  of  the  cylinder.  An  electric  spark  ignites  the 
explosive  mixture,  and  the  force  of  the  explosion  drives  the 
piston  violently  to  the  right  (see  3).  At  the  beginning  of 
the  return  stroke  (see  4)  the  exhaust  valve  D  opens,  and 
as  the  piston  moves  to  the  left,  the  spent  gaseous  products  of 
the  explosion  are  forced  out  of  the  cylinder.  The  initial 
condition  is  thus  restored  and  the  cycle  begins  over  again. 


FIG.  189.    Principle  of  the  gas 
engine 


INDUSTRIAL  APPLICATIONS 


191 


Since  it  is  only  during  the  third  stroke  that  the  engine  is 
receiving  energy  fronKthe  exploding  gas,  the  flywheel  is 
always  made  very  heavy  so  that  the  energy  stored  up  in  it  in 
the  third  stroke  may  keep  the  machine  running  with  little  loss 
of  speed  during  the  other  three  parts  of  the  cycle. 

251.  Mechanism  of  the  gas  engine.  The  mechanism  by  which  the 
above  operations  are  carried  out  in  one  type  of  modern  gas  engine 
may  be  seen  from  a  study  of  Figs.  190  and  191.  Fig.  191  is  a  section  of  the 


FIG.  190.   The  gas  engine 


left  end  of  the  engine  shown  in  perspective  in  Fig.  190.  Suppose  that  the 
flywheels  W  are  first  set  in  motion  by  hand.  When  the  cam,  or  eccentric  ct 
(Fig.  190),  drives  the.rod.ft  to  the  left,  it  opens  a  valve  in  .F  through  which 
gas  passes  from  the  inlet  pipe  A  into  the  mixing  chamber  /  (Figs.  190  and 
191).  Here  it  mixes  with  air  which  entered  through  the  pipe  B.  As  soon 
as  the  cam  c2  has  moved  about  to  the  position  in  which  it  throws  the  lever 
arm  ^  to  the  left,  the  rod  Gl  is  forced  upward  and  the  inlet  valve  E  (Fig.  191) 
is  therefore  opened.  This  happens  at  the  beginning  of  stage  1  (§  250) 
when  the  piston  K  is  beginning  to  move  to  the  right.  Hence  the  explo- 
sive mixture  is  at  once  drawn  into  C*(Fig.  191).  At  the  beginning  of  stage 
No.  3  a  third  eccentric  rod  N  operated  by  an  eccentric  c4  (Fig.  190)  breaks 
t 


192 


WORK  AND  HEAT  ENERGY 


EIG.  191.    Section  through  end  of 
gas  engine 


an  electric  contact  at  i  (Fig.  191),  and  thus  produces  a  spark  which  ex- 
plodes the  gas.  At  the  beginning  of  stage  4  the  cam  cs  drives  the  lever 
arm  /2  (Fig.  190)  to  the  left,  and  thus 
with  the  aid  of  £2  (Figs.  190  and  191) 
opens  the  exhaust  valve  D  (Fig.  191) 
and  thus  permits  the  spent  gases  to 
escape.  This  completes  the  cycle. 

Since  each  of  the  four  cams,  cv  c2, 
c'3,  c4,  must  open  its  valve  once  in  two 
revolutions  of  the  flywheel,  all  four 
of  these  cams  are  placed  not  on  the 
main  shaft  H,  but  on  the  axle  of  the 
gear  wheel  M,  which  has  twice  as 
many  teeth  as  has  the  gear  wheel  n 
on  the  main  shaft.  M  therefore  re- 
volves but  once  while  the  main  shaft 
is  revolving  twice.  In  order  that  the 
cylinder  may  be  kept  cool,  it  is  sur- 
rounded by  a  jacket  U through  which  water  is  kept  continually  circulating. 

The  efficiency  of  the  gas  engine  is  often  as  high  as  25%,  or  nearly 
double  that  of  the  best  steam  engines.  Furthermore,  it  is  free  from 
smoke,  is  very  compact,  and  may 
be  started  at  a  moment's  notice. 
On  the  other  hand,  the  fuel,  gas 
or  gasoline,  is  comparatively  ex- 
pensive. Most  automobiles  are 
run  by  gasoline  engines,  chiefly 
because  the  lightness  of  the  en- 
gine and  of  the  fuel  to  be  car- 
ried are  here  considerations  of 
great  importance. 

It  has  been  the  development 
of  the  light  and  efficient  gas  en- 
gine which  has  made  possible 
man's  recent  conquest  of  the  air 
through  the  use  of  the  aeroplane 
and  airship. 

252.  The  steam  turbine.  The 
steam  turbine  represents  the  latest  development  of  the  heat  engine.  In 
principle  it  is  very  much  like  the  common  windmill,  the  chief  difference 
being  that  it  is  steam  instead  of  air  which  is  driven  at  a  high  velocity 


FIG.  192.   The  principle  of  the  steam 
turbine 


INDUSTRIAL  APPLICATIONS 


193 


against  a  series  of  blades  which  are  arranged  radially  about  the  cir- 
cumference of  the  wheel  which  is  to  be  set  into  rotation.  The  steam, 
however,  unlike  the  wind,  is  always  directed  by  nozzles  at  the  angle  of 
greatest  efficiency  against  the  blades  (see  Fig.  192).  Furthermore,  since 
the  energy  of  the  steam  is  not  nearly  spent  after  it  has  passed  through 
one  set  of  blades,  such  as  that  shown  in  Fig.  192,  it  is  in  practice  always 
passed  through  .a  whole  series  of  such  sets  (Fig.  193),  every  alternate 

Exhaust 


Revolving 


Stationary 


Revolving 


Nozzle 


FIG.  193.    Path  of  steam  in  Curtis's  turbine 


row  of  which  is  rigidly  attached  to  the  rotating  shaft,  while  the  inter- 
mediate rows  are  fastened  to  the  immovable  outer  jacket  of  the  engine, 
and  only  serve  as  guides  to  redirect  the  steam  at  the  most  favorable 
angle  against  the  next  row  of  movable  blades.  In  this  way  the  steam  is 
kept  alternately  bounding  from  fixed  to  movable  blades  till  its  energy  is 
expended.  The  number  of  rows  of  blades  is  often  as  high  as  sixteen. 

Turbines  are  at  present  coming  rapidly  into  use,  chiefly  for  large- 
power  purposes.  Their  advantages  over  the  reciprocating  steam  engine 
lie  first  in  the  fact  that  they  run  with  almost  no  jarring,  and  therefore 


194 


WORK  AND  HEAT  ENERGY 


Air  at  200  atmospheres^ 


require  much  lighter  and  less  expensive  foundations;  and  second,  in  the 
fact  that  they  occupy  less  than  one  tenth  the  floor  space  of  ordinary 
engines  of  the  same  capacity.  Their  efficiency  is  fully  as  high  as  that 
of  the  best  reciprocating  engines.  The  highest  speeds  attained  by  ves- 
sels at  sea,  namely  about  40  miles  per  hour,  have  been  made  with  the 
aid  of  steam  turbines.  The  largest  vessel  which  has  thus  far  ever  been 
launched,  the  Hamburg-American  steamer  Imperator,  919  feet  long,  98 
feet  wide,  100  feet  high  (from  the  keel  to  the  top  of  her  ninth  deck), 
having  a  total  "  displacement "  of  70,000  tons  and  a  speed  of  22J  knots, 
is  driven  by  four  steam  turbines  having  a  total  horse  power  of  72,000. 
One  of  the  immense  rotors  contains  50,000  blades  and  develops  22,000 
horse  power. 

253.  The  liquid-air  machine.    In  the  actual  manufacture  of  liquid 
air  a  pressure  pump  P  (Fig.  194)  forces  the  air  into  a  spiral  coil  C 
under  a  pressure  of  about  200  atmospheres.    The  heat  produced  by  this 
compression  is  carried  off  by  running  water  which  circulates  through 
the  tank  JR.    The  cock  c  is  then 

opened  and  the  air  expands  from 
200  atmospheres  down  to  1  atmos- 
phere. In  this  expansion  the  tem- 
perature falls.  This  cooled  air 
returns  through  a  larger  spiral  S 
which  incloses  the  high-pressure 
spiral  s,  and  thus  cools  off  the  air 
which  is  coming  down  to  the  ex- 
pansion valve  through  the  inner 
spiral.  In  this  process  the  tem- 
perature of  the  air  issuing  from 
the  valve  c  continuously  falls  un- 
til it  reaches  the  temperature  of 
liquefaction.  Liquid  air  can  then 

be  drawn  off  through  the  stopcock  R.  The  air  which  escapes  lique- 
faction returns  to  the  compressor,  where  it  is  again  forced  into  the 
inner  spiral  s,  together  with  a  certain  amount  of  air  which  enters  from 
the  outside  at  o. 

254.  Manufactured  ice.    In  the  great  majority  of  modern  ice  plants 
the  low  temperature  required  for  the  manufacture  of  the  ice  is  produced 
by  the  rapid  evaporation  of  liquid  ammonia.   At  ordinary  temperatures 
ammonia  is  a  gas,  but  it  may  be  liquefied  by  pressure  alone.    At  80°  F. 
a  pressure  of  155  pounds  per  square  inch,  or  about  10  atmospheres,  is 
required  to  produce  its  liquefaction.    Fig.  195  shows  the  essential  parts 


FIG.  194.   The  liquefaction  of  air 


H  3 


INDUSTRIAL  APPLICATIONS 


195 


of  an  ice  plant.  The  compressor,  which  is  usually  run  by  a  steam  engine, 
forces  the  gaseous  ammonia  under  a  pressure  of  155  pounds  into  the  con- 
denser coils  shown  on  the  right,  and  there  liquefies  it.  The  heat  of  con- 
densation of  the  ammonia  is  carried  off  by  the  running  water  which 
constantly  circulates  about  the  condenser  coils.  From  the  condenser 
the  liquid  ammonia  is  allowed  to  pass  very  slowly  through  the  regulat- 
ing valve  V  into  the  coils  of  the  evaporator,  from  which  the  evaporated 
ammonia  is  pumped  out  so  rapidly  that  the  pressure  within  the  coils 
does  not  rise  above  34  pounds.  It  will  be  noted  from  the  figure  that 
the  same  pump  which  is  there  labeled  the  compressor  exhausts  the 


Low  Pressure 
Gau 


HighJPressure 
Gauge 


FIG.  195.  Compression  system  of  ice  manufacture 

ammonia  from  the  evaporating  coils  and  compresses  it  in  the  condensing 
coils ;  for,  just  as  in  Fig.  194,  the  valves  are  so  arranged  that  the  pump 
acts  as  an  exhaust  pump  on  one  side  and  as  a  compression  pump  on  the 
other.  The  rapid  evaporation  of  the  liquid  ammonia  under  the  reduced 
pressure  existing  within  the  evaporator  cools  these  coils  to  a  temperature 
of  about  5°  F.  for  every  gram  which  evaporates.  The  brine  with  which 
these  coils  are  surrounded  has  its  temperature  thus  reduced  to  about  16° 
or  18°  F.  This  brine  is  made  to  circulate  about  the  cans  containing 
the  water  to  be  frozen. 

255.  Cold  storage.  The  artificial  cooling  of  factories  and  cold-storage 
rooms  is  accomplished  in  a  manner  exactly  similar  to  that  employed 
in  the  manufacture  of  ice.  The  brine  is  cooled  exactly  as  described 
above,  and  is  then  pumped  through  coils  placed  in  the  rooms  to  be 


196  WORK  AND  HEAT  ENERGY 

cooled.  In  some  systems  carbon  dioxide  is  used  in  place  of  ammonia, 
but  the  principle  is  in  no  way  altered.  Sometimes,  too,  the  brine  is 
dispensed  with,  and  the  air  of  the  rooms  to  be  cooled  is  forced  by  means 
of  fans  directly  over  the  cold  coils  containing  the  evaporating  ammonia 
or  carbon  dioxide.  It  is  in  this  way  that  theaters  and  hotels  are  cooled. 

QUESTIONS  AND  PROBLEMS 

1.  Why  is  not  the  boiling  point  of  water  in  the  boiler  of  a  steam 
engine  100°  C.  ? 

2.  If  the  average  pressure  in  the  cylinder  of  a  steam  engine  is  10 
kilos  to  the  square  centimeter,  and  the  area  of  the  piston  is  427  sq.  cm., 
how  much  work  is  done  by  the  piston  in  a  stroke  of  length  50  cm.? 
How  many  calories  did  the  steam  lose  in  this  operation  ? 

3.  The  total  efficiency  of  a  certain  600  H.P.  locomotive  is  6% ;  8000 
calories  of  heat  are  produced  by  the  burning  of  1  g.  of  the  best  an- 
thracite coal;  how  many  kilos  of  such  coal  are  consumed  per  hour 
by  this  engine  ?    (Take  1  H.P.  =  746  watts  and  1  calorie  per  second 
4.2  watts.) 

4.  It  requires  a  force  of  300  kilos  to  drive  a  given  boat  at  a  speed  of 
15  knots  (25  km.).    How  much  coal  will  be  required  to  run  this  boat  at 
this  speed  across  a  lake  300  km.  wide,  the  efficiency  of  the  engines  being 
7  %  and  the  coal  being  of  a  grade  to  furnish  6000  calories  per  gram  ? 

5.  What  total  pushing  force  do  the  propellers  of  the  Lusitania  exert 
when  she  is  using  her  maximum  horse  power  (70,000)  and  is  running 
at  25  knots  (46.4  km.)  per  hour? 

6.  The  best  triple-expansion  marine  engines  consume  1.5  Ib.  of  coal 
per  hour  per  H.P.,  or  .91  kg.  per  indicated  kilowatt  (see  Ency.  Brit., 
"  Steam  Engine  ").    If  the  coal  furnishes  6000  calories  per  gram,  what 
is  the  efficiency  ? 

7.  If  liquid  air  is  placed  in  an  open  vessel,  its  temperature  will  not 
rise  above  —  182°  C.   Why  not?   Suggest  a  way  in  which  its  temperature 
could  be  made  to  rise  above  —  182°  C.,  and  a  way  in  which  it  could  be 
made  to  fall  below  that  temperature. 

8.  The  average  locomotive  has  an  efficiency  of  about  6  %.    What 
horse  power  does  it  develop  when  it  is  consuming  1  ton  of  coal  per 
hour?    (See  Problem  6,  above.) 

9.  What  pull  does  a  1000  H.P.  locomotive  exert  when  it  is  running 
at  25  miles  per  hour  and  exerting  its  full  horse  power? 


CHAPTER  X 

THE   TRANSFERENCE  OF  HEAT 

CONDUCTION 

256.  Conduction  in  solids.  If  one  end  of  a  short  metal  bar  be 
held  in  the  fire,  the  other  end  soon  becomes  too  hot  to  hold.  But  if  the 
metal  rod  is  replaced  by  one  of  wood  or  glass,  the  end  away  from  the 
flame  will  not  be  appreciably  heated. 

This  experiment  and  others  like^'it  show  that  nonmetallic 
substances  possess  a  much  smaller  ability  to  conduct  heat  than 
do  metallic  substances.  But  although 
all  metals  are  good  conductors  as  com- 
pared with  nonmetals,  they  differ  widely 
among  themselves  in  their  conducting 
powers. 

Let  copper,  iron,  and  German  silver  wires 
50  cm.  long  and  about  3  mm.  in  diameter  be      ^  ^  ~Differences  in 
twisted  together  at  one  end  as  in  Fig.  196,  conductivities     of 

and  let  a  Bunsen  flame  be  applied  to  the  metals 

twisted  ends.    Let  a  match  be   slid   slowly 

from  the  cool  end  of  each  wire  toward  the  hot  end,  until  the  heat  from 
the  wire  ignites  it.  The  copper  will  be  found  to  be  the  best  conductor 
and  the  German  silver  the  poorest. 

In  the  following  table  some  common  substances  are  arranged 
in  the  order  of  their  heat  conductivities.  The  measurements 
have  been  made  by  a  method  not  differing  in  principle  from 
that  just  described.  For  convenience,  silver  is  taken  as  100. 


Silver 

100 

Tin 

15 

Mercury      .    . 

.    1.35 

Copper 

.    .      74 

Iron 

12 

Ice       .... 

.21 

Gold 

.    .      53 

L/ead      .    .    . 

8.5 

Glass       .    .    . 

.      .046 

Brass 

.    .      27 

German  silver  .    . 
197 

6.3 

Hard  rubber   . 

.      .024 

198 


THE  TRANSFEKENCE  OF  HEAT 


FIG.  197.    Water  a  nonconductor 


257.  Conduction  in  liquids  and  gases.    Let  a  small  piece  of  ice 
be  heldtby  means  of  a  glass  rod  in  the  bottom  of  a  test  tube  full  of 
ice  water.    Let  the  upper  part  of 

the  tube  be  heated  with  a  Bunsen 
burner  as  in  Fig..  197.  The  upper 
part  of  the  water  may  be  boiled  for 
some  time  without  melting  the  icej 
Water  is  evidently,  then,  a  very  poor 
conductor  of  heat.  The  same  thing 
may  be  shown  more  strikingly  as 
follows :  The  bulb  of  an  air  ther- 
mometer is  placed  only  a  few  milli- 
meters beneath  the  surface  of  water 
contained  in  a  large  funnel  arranged 
as  in  Fig.  198.  If,  now,  a  spoonful  of 

ether  is  poured  on  the  water  and  set  on  fire,  the  index  of  the  air  ther- 
mometer will  show  scarcely  any  change,  in  spite  of  the  fact  that  the  air 
thermometer  is  a  very  sensitive  indicator  of  changes  in  temperature. 

Careful  measurements  of  the  conductivity 
of  water  show  that  it  is  only  about  121OQ  of 
that  of  silver.  The  conductivity  of  gases  is 
even  smaller,  not  amounting  on  the  average 
to  more  than  -^  that  of  water. 

258.  Conductivity  and  sensation.    It  is  a 
fact  of  common  observation  that  on  a  cold 
day  in  winter  a  piece  of  metal  feels  much 
colder  to  the  hand  than  a  piece  of  wood, 
notwithstanding  the  fact  that  the  temper- 
ature  of  the   wood  must  be  the   same   as 
that    of    the    metal.     On    the    other   hand, 

if  the  same  two  bodies  had  been  lying  in    ether  on  the  water 

the  hot  sun  in  midsummer,  the  wood  might    does  ^  affect  the 

'.  °  air  thermometer 

be    handled    without    discomfort,    but    the 

metal  would  be  uncomfortably  hot.  The  explanation  of 
these  phenomena  is  found  in  the  fact  that  the  iron,  being 
a  much  better  conductor  than  the  wood,  removes  heat  from 


t.  198.    Burning 


CONDUCTION  199 

X^ 

the  hand  much  more  rapidly  in  winter,  and  imparts  heat  to  the 
hand  much  more  rapidly  in  summer,  than  does  the  wood.  In 
general,  the  better  a  conductor  the  hotter  it  will  feel  to  a 
hand  colder  than  itself,  and  the  colder  to  ji__hand  hotter  than 
itsejf.  Thus  in  a  cold  room  oilcloth,  a  fairly  good  conductor, 
feels  much  colder  to  the  touch  than  a  carpet,  a  comparatively 
poor  conductor.  For  the  same  reason  linen  clothing  feels 
cooler  to  the  touch  in  winter  than  woolen  goods. 

259.  The  role  of  air  in  nonconductors.    Feathers,  fur,  felt, 
etc.,  make  very  warm  coverings,  because  they  are  very  poor 
conductors  of  heat  and  thus  prevent  the  escape  of  heat  from 
the  bodyrTheir  poor  conductivity  is  due  in  large  measure  to 
the  fact  that  they  are  full  of  minute  spaces  containing  air,  and 
gases  are  the  best  nonconductors  of  heat.    It  is  for  this  reason 
that  freshly  fallen  snow  is  such  an  efficient  protection  to  vege- 
tation.   Farmers  always  fear  for  their  fruit  trees  and  vines 
when  there  is  a  severe  cold  snap  in  winter,  unless  there  is  a 
coating  of  snow  on  the  ground  to  prevent  a  deep  freezing. 

260.  The  Davy  safety  lamp.    Let  a  piece  of  wire  gauze  be  held 
above  an  open  gas  jet,  and  a  match  applied  above  the  gauze.  The  flame 
will  be  found  to  burn  above  the  gauze 

as  in  Fig.  199,  (1),  but  it  will  not 
pass  through  to  the  lower  side.  If 
it  is  ignited  below  the  gauze,  the 
flame  will  not  pass  through  to  the 
upper  side  but  will  burn  as  shown 
in  Fig.  200,  (2). 

The  explanation  is  found  in      FIG.  199.  A  flame  will  not  pass 
,  i      j.     ,   ,1     ,  ,  i  i  through  wire  gauze 

the  fact  that  the  gauze  conducts 

the  heat  away  from  the  flame  so  rapidly  that  the  gas  on  the 
other  side  is  not  raised  to  the  temperature  of  ignition.  Safety 
lamps  used  by  miners  are  completely  incased  in  gauze,  so  that 
if  the  mine  is  full  of  inflammable  gases,  they  are  not  ignited 
by  the  lamp  outside  of  the  gauze. 


200  THE  TRANSFERENCE  OF  HEAT 

QUESTIONS  AND  PROBLEMS 

1.  Why  is  the  outer  pail  of  an  ice-cream  freezer  made  of  thick  wood 
and  the  inner  can  of  thin  metal  ? 

2.  Why  do  firemen  wear  flannel  shirts  in  summer  to  keep  cool  and 
in  winter  to  keep  warm  ? 

3.  Why  do  we  wrap  up  ice  cream  in  thick  woolen  blankets  in  sum- 
mer to  keep  it  from  melting  ? 

4.  If  ice  in  a  refrigerator  is  wrapped  up  in  blankets,  what  is  the  effect 
on  the  ice  ?  on  the  refrigerator  ? 

5.  If  a  piece  of  paper  is  wrapped  tightly  around  a  metal  rod  and 
held  for  an  instant  in  a  Bunsen  flame,  it  will  not  be  scorched.    If  held 
in  a  flame  when  wrapped  around  a  wooden  rod,  it  will  be  scorched  at 
once.    Explain. 

6.  If  one  touches  the  pan  containing  a  loaf  of  bread  in  a  hot  oven, 
he  receives  a  much  more  severe  burn  than  if  he  touches  the  bread  itself, 
although  the  two  are  at  the  same  temperature.    Explain. 

7.  Why  are  plants  often  covered  with  paper  on  a  night  when  frost 
is  expected  ? 

8.  Why  will  a  moistened  finger  or  the  tongue  freeze  instantly  to  a 
piece  of  iron  on  a  cold  winter's  day,  but  not  to  a  piece  of  wood  ? 

9.  Does  clothing  ever  afford  us  heat  in  winter?    How,  then,  does  it 
keep  us  warm  ? 

CONVECTION 

261.  Convection  in  liquids.  Although  the  conducting  power 
of  liquids  is  so  small,,  as  was  shown  in  the  experiment  of  §  257, 
they  are  yet  able,  under  certain  circumstances,  to  transmit 
heat  much  more  effectively  than  solids.  Thus,  if  the  ice  in  the 
experiment  of  Fig.  197  had  been  placed  at  the  top  and  the 
flame  at  the  bottom,  the  ice  would  have  been  melted  very 
quickly.  This  shows  that  heat  is  transferred  with  enormously 
greater  readiness  from  the  bottom  of  the  tube  toward  the  top 
than  from  the  top  toward  the  bottom.  The  mechanism  of 
this  heat  transference  will  be  evident  from  the  following 
experiment : 

Let  a  round-bottomed  flask  (Fig.  200)  be  half  filled  with  water  and 
a  few  crystals  of  magenta  dropped  into  it.  Then  let  the  bottom  of  the 
flask  be  heated  with  a  Bunsen  burner.  The  magenta  will  reveal  the  fact 


201 


FIG.  200.  Convec- 
tion currents 


that  the  heat  sets  up  currents  the  direction  of  which  is  upward  in  the 
region  immediately  above  the  flame  but  downward  at  the  sides  of 
the  vessel.  It  will  not  be  long  before  the  whole  of 
the  water  is  uniformly  colored.  This  shows  how 
thorough  is  the  mixing  accomplished  by  the  heating. 

The  explanation  of  the  phenomenon  is  as 
follows :  The  water  nearest  the  flame  became 
heated  and  expanded.  It  was  thus  rendered 
less  dense  than  the  surrounding  water,  and 
was  therefore  forced  to  the  top  by  the  pres- 
sure transmitted  from  the  colder  and  there- 
fore denser  water  at  the  sides  which  then 
came  in  to  take  its  place. 

It  is  obvious  that  this  method  of  heat  trans- 
fer is  applicable  only  to  fluids.  The  essential 
difference  between  it  and  conduction  is  that 
the  heat  is  not  transferred  from  molecule  to  molecule  through- 
out the  whole  mass,  but  is  rather  transferred  by  the  bodily 
movement  of  comparatively  large  masses  of  the  heated  liquid 
from  one  point  to  another.  This  method  of  heat  transference 
is  known  as  convection. 

262.  Winds  and  ocean  currents.  Winds  are  convection  cur- 
rents in  the  atmosphere  caused  by  unequal  heating  of  the 
earth  by  the  sun.  Let  us  consider,  for  example,  the  land  and 
sea  breezes  so  familiar  to  all  dwellers  near  the  coasts  of  large 
bodies  of  water.  During  the  daytime  the  land  is  heated 
more  rapidly  than  the  sea,  because  the  specific  heat  of  water  is 
much  greater  than  that  of  eajn.  Hence  the  hot  air  over  the 
land  expands  and'  is  forced 'up  by  the  colder  and  denser  air 
over  the  sea  which  moves  in  to  take  its  place.  This  constitutes 
the  sea  breeze  which  blows  during  the  daytime,  usually  reach- 
ing its  maximum  strength  in  the  late  afternoon.  At  night  the 
earth  cools  more  rapidly  than  the  sea  and  hence  the  direction 
of  the  wind  is  reversed.  The  effect  of  these  breezes  is  seldom 
felt  more  than  twenty-five  miles  from  shore. 


202  THE  TRANSFERENCE  OF  HEAT 

Ocean  currents  are  caused  partly  by  the  unequal  heating  of 
the  sea  and  partly  by  the  direction  of  the  prevailing  winds. 
In  general  both  winds  and  currents  are  so  modified  by  the  con- 
figuration of  the  continents  that  it  is  onjy  over  broad  expanses 
of  the  ocean  that  the  direction  of  either  can  be  predicted  from 
simple  considerations. 

RADIATION 

263.  A  third  method  of  heat  transference.   There  are  certain 
phenomena  in  connection  with  the  transfer  of  heat  for  which 
conduction  and  convection  .are  wholly  unable  to  account.    For 
example,  if  one  sits  in  front  of  a  hot  grate  fire,  the  heat  which 
he  feels  cannot  come  from  the  fire  by  convection,  because  the 
currents  of  air  are  moving  toward  the  fire  rather  than  away 
from  it.    It  cannot  be  due  to  conduction,  because  the  con- 
ductivity of  air  is  extremely  small  and  the  colder  currents  of 
air  moving  toward  the  fire  would  more  than  neutralize  any 
transfer  outward  due  to  conduction.    There  must  therefore  be 
some  way  in  which  heat  travels  across  the  intervening  space 
other  than  by  conduction  or  convection. 

It  is  still  more  evident  that  there  must  be  a  third  method 
of  heat  transfer  when  we  consider  the  heat  which  comes  to 
us  from  the  sun.  Conduction  and  convection  take  place  only 
through  the  agency  of  matter;  but  we  know  that  the  space 
between  the  earth  and  the  sun  is  not  filled  with  ordinary  mat- 
ter, or  else  the  earth  would  be  retarded  in  its  motion  through 
space.  Radiation  is  the  name  given  to  this  third  method  by 
which  heat  travels  from  one  place  to  another,  and  which  is 
illustrated  in  the  passing  of  heat  from  a  grate  fire  to  a  body 
in  front  of  it,  or  from  the  sun  to  the  earth. 

264.  The  nature  of  radiation.  The  nature  of  radiation  will 
be  discussed  more  fully  in  Chapter  XXI.    It  will  be  suffi- 
cient here  to  call  attention  to  the  following  differences  between 
conduction,  convection,  and  radiation. 


HEATING  AND  VENTILATING  203 

First,  while  conduction  and  convection  are  comparatively 
slow  processes,  the  transfer  of  heat  by  radiation  takes  place 
with  the  enormous  speed  with  which  light  travels,  namely 
186,000  miles  per  second.  That  the  two  speeds  are  the  same 
is  evident  from  the  fact  that  at  the  time  of  an  eclipse  of  the 
sun  the  shutting  off  of  heat  from  the  earth  is  observed  to  take 
place  at  the  same  time  as  the  shutting  off  of  light. 

Second,  radiant  heat  travels  in  straight  lines,  while  conducted 
or  convected  heat  may  follow  the  most  circuitous  routes.  The 
proof  of  this  statement  is  found  in  the  familiar  fact  that  ra- 
diation may  be  cut  off  by  means  of  a  screen  placed  directly 
between  a  source  and  the  body  to  be  protected. 

Third,  radiant  heat  may  pass  through  a  medium  without 
heating  it.  This  is  shown  by  the  fact  that  the  upper  regions 
of  the  atmosphere  are  very  cold,  even  in  the  hottest  days  in 
summer,  or  that  a  hothouse  may  be  much  warmer  than  the 
glass  through  which  the  sun's  rays  enter  it. 

THE  HEATING  AND  VENTILATING  OP  BUILDINGS 

265.  The  principle  of  ventilation.   The 

heating  and  ventilating  of  buildings  are 
accomplished  chiefly  through  the  agency 
of  convection. 

To  illustrate  the  principle  of  ventilation,  let  a  candle 
be  lighted  and  placed  in  a  vessel  containing  a  layer  of 
water  (Fig.  201).  When  a  lamp  chimney  is  placed  over 
the  candle  so  that  the  bottom  of  the  chimney  is  under 
the  water,  the  flame  will  slowly  die  down  and  will 
finally  be  extinguished.  This  is  because  the  oxygen, 
which  is  essential  to  combustion,  is  gradually  used  up 
and  no  fresh  supply  is  possible  with  the  arrangement 
described.  If  the  chimney  is  raised  even  a  very  little 
above  the  water,  the  dying  flame  will  at  once  brighten.  ^IG  201  Con- 
Why?  If  a  metal  or  cardboard  partition  is  inserted  vection  currents 
in  the  chimney,  as  in  Fig.  201,  the  flame  will  burn  in  air 


204 


THE  TRANSFERENCE  OF  HEAT 


continuously,  even  when  the  bottom  of  the  chimney  is  under  water.  The 
reason  will  be  clear  if  a  piece  of  burning  touch  paper  (blotting  paper 
soaked  in  a  solution  of  potassium  nitrate  and  dried)  is  held  over  the 
chimney.  The  smoke  will  show  the  direction  of  the  air  currents.  If  the 
chimney  is  a  large  one,  in  order  that  the  first  part  of  the  above  experi- 
ment may  succeed,  it  may  be  necessary  to  use  two  candles  ;  for  too  small 
a  heated  area  permits  the  formation  of  downward  currents  at  the  sides. 

266.  Ventilation  of  houses.    In  order  to  secure  satisfactory 
ventilation  it  is  estimated  that  a  room  should  be  supplied  with 
2000  cubic  feet  of  fresh  air  per  hour  for  each  occupant  (a  gas 
burner  is  equivalent  in  oxygen  consumption  to  four  persons). 
A  current  of  air  moving  with  a  speed  great  enough  to  be  just 
perceptible  has  a  velocity  of  about  3  feet  per  second.  Hence  the 

area  of  opening  required  for  each  person  when  fresh  air 
is  entering  at  this  speed  is  about  25  or  30  square  inches. 

The  manner  of  sup- 
plying this  requisite 
amount  of  fresh  air 
in  dwelling  houses 
depends  upon  the 
method  of  heating 
employed. 

If  a  house  is 
heated  by  stoves  or 
fireplaces,  no  special 
provision  for  ven- 
tilation is  needed. 
The  foul  air  is 
drawn  up  the  chim- 
F,o.  202.  Hot-air  heating  ney  with  the  smoke, 

and  the  fresh  air  which  replaces  it  finds  entrance  through 
cracks  about  the  doors  and  windows  and  through  the  walls. 

267.  Hot-air  heating.   In  houses  heated  by  hot-air  furnaces  an  air 
duct  ought  always  to  be  supplied  for  the  entrance  of  fresh  cold  air,  in 
the  manner  shown  in  Fig.  202  (see  "cold-air  inlet").    This  cold  air 


HEATING  AND  VENTILATING 


205 


FIG.  203.    Princi- 


from  out  of  doors  is  heated  by  passing  in  a  circuitous  way,  as  shown 
by  the  arrows,  over  the  outer  jacket  of  iron  which  covers  the  fire  box. 
It  is  then  delivered  to  the  rooms.  Here  a  part  of  it  escapes  through 
windows  and  doors,  and  the  rest  returns  through  the 
cold-air  register  to  be  reheated,  after  being  mixed 
with  a  fresh  supply  from  out  of  doors. 

The  course  of  the  air  which  feeds  the  fire  is  shown 
by  the  dotted  arrows.  When  the  fire  is  first  started, 
in  order  to  gain  a  strong  draft  the  damper  C  is 
opened  so  that  the  smoke  may  pass  directly  up  the 
chimney.  After  the  fire  is  under  way  the  damper  C 
is  closed  so  that  the  smoke  and  hot  gases  from  the 
furnace  nrnst  pass,  as  in- 
dicated by  the  arrows, 
over  a  roundabout  path, 
in  the  course  of  which 

they  give  up  the  major 
pie  of  hot-water.          t   of   their    heat   to 
heating 


the  steel  walls  of  the 
jacket,  which  in  turn  pass  it  on  to  the  air 
which  is  on  its  way  to  the  living  rooms. 

268.  Hot-water  heating.  To  il- 
lustrate the  principle  of  hot-water 
heating  let  the  arrangement  shown 
in 'Fig.  203  be  set  up,  the  upper 
vessel  being  filled  with  colored 
water,  and  then  let  a  flame  be 
applied  to  the  lower  vessel.  The 
colored  water  will  show  that  the 
current  moves  in  the  direction  of 
the  arrows. 

The  actual  arrangement  of  boiler 
and  radiators  in  one  system  of  hot- 
water  heating  is  shown  in  Fig.  204. 
The  water  heated  in  the  furnace 
rises  directly  through  the  pipe  A  to 
a  radiator  R,  and  returns  again  to 
the  bottom  of  the  furnace  through 
the  pipes  B  and  D.  The  circula- 


Fresli- 
Air— 
Inlet 


FIG.  204.    Hot-water  heater 

tion  is  maintained  because  the  column  of  water  in  A  is  hotter  and 
therefore   lighter  than  the  water  in  the  return  pipe  B. 


206 


THE  TRANSFERENCE  OF  HEAT 


In  the  most  common  system  of  hot-water  or  steam  heating,  the 
so-called  direct-radiation  system,  no  provision  whatever  is  made  for 
ventilation.  The  occupants  must  de- 
pend entirely  on  open  windows  for  their 
supply  of  fresh  air.  In  the  so-called 
direct-indirect  system,  shown  in  Fig.  204, 
fresh  air  is  introduced  through  the  radi- 
ator itself.  The  indirect  system  differs 
from  this  only  in  that  steam  or  hot- 
water  coils,  instead  of  being  in  the  rooms, 
are  suspended  from  the  ceiling  of  the 
basement  in  wooden  boxes  (Fig.  205). 
The  arrows  indicate  the  direction  which 
the  air  currents  take  as  they  pass  from  out 
of  doors,  through  the  heating  coils,  and 
finally  through  the  register  into  the  room.  FIG.  205.  Indirect  system 


QUESTIONS  AND  PROBLEMS 

1.  If  2  metric  tons  of  coal  are  burned  per  month  in  your  house 
and  if  your  furnace  allows  one  third  of  the  heat  to  go  up  the  chim- 
ney, how  many  calories  do  you  use  per  day  ?    (Take  1  g.  as  yielding 
6000  calories.) 

2.  Explain  the  underlying  principles  of  the  fireless  cooker. 

3.  Why  is  a  hollow  wall  filled  with  sawdust  a  better  nonconductor 
of  heat  than  the  same  wall  filled  with  air  alone? 

4.  In  a  system  of  hot-water  heating  why  does  the  return  pipe  always 
connect  at  the  bottom  of  the  boiler,  while  the  outgoing  pipe  connects 
with  the  top  ? 

5.  Which  is  thermally  more  efficient,  a  cook  stove  or  a  grate?  Why  ? 

6.  When  a  room  is  heated  by  a  fireplace,  which  of  the  three  methods 
of  heat  transference  plays  the  most  important  r61e  ? 

7.  Which  methods  of  heat  transfer  are  most  important  in  systems 
of  direct  and  of  indirect  radiation  ? 

8.  Why  do  you  blow  on  your  hands  to  warm  them  in  winter  and 
fan  yourself  for  coolness  in  summer? 

9.  If  you  open  a  door  between  a  warm  and  a  cold  room,  in  what 
direction  will  a  candle  flame  be  blown  which  is  placed  at  the  top  of  the 
door  ?   Explain. 

10.  Why  is  felt  a  better  conductor  of  heat  when  it  is  very  firmly 
packed  than  when  loosely  packed  ? 


CHAPTER  XI 

MAGNETISM* 
GENERAL  PROPERTIES  OF  MAGNETS 

269.  Magnets.  It  has  been  known  for  many  centuries  that 
some  specimens  of  the  ore  known  as  magnetite  (Fe3O4)  have 
the  property  of  attracting  small  bits  of  iron  and  steel.  This 
ore  probably  received  its  name  from  the  fact  that  it  is  espe- 
cially abundant  in  the  province  of  Magnesia,  in  Thessaly, 
although  the  Latin  writer  Pliny  says  that  the  word  "  magnet " 
is  derived  from  the  name  of  the  Greek  shepherd  Magnes,  who, 
on  the  top  of  Mount  Ida,  observed  the  attraction  of  a  large 
stone  for  his  iron  crook.  Pieces  of  this  ore  which  exhibit  this 
attractive  property  are  known  as  natural  magnets. 

It  was  also  known  to  the  ancients  that  artificial  magnets 
may  be  made  by  stroking  pieces  of  steel  with  natural  magnets, 
but  it  was  not  until  about  the  twelfth  century  that  the  dis- 
covery was  made  that  a  suspended  magnet  will  assume  a  north- 
and-sonth  position.  Because  of  this  latter  property  natural 
magnets  became  known  as  lodestones  (leading  stones),  and 
magnets,  either  artificial  or  natural,  began  to  be  used  for 
determining  directions.  The  first  mention  of  the  use  of  the 
compass  in  Europe  is  in  1190.  It  is  thought  to  have  been 
introduced  from  China. 

Magnets  are  now  made  either  by  stroking  bars  of  steel  in 
one  djj^cticoijwitli_a_magnet,  or  by  passing  electric  currents 

*  This  chapter  should  either  be  accompanied  or  preceded  by  laboratory  experi- 
ments on  magnetic  fields  and  on  the  molecular  nature  of  magnetism.  See,  for 
example,  Experiments  25  and  26  of  the  authors'  manual. 

t  207 


208 


MAGNETISM 


about  the  bars  in  a  manner  to  be  described  later.  The  form 
shown  in  Fig.  206  is  called  a  bar  magnet,  that  shown  in 
Fig.  207  a  horseshoe  magnet.  The  lat-  _ 

ter  form  is  the  more  common. 

TJ,  .  .     ,.          T  .    ,      .          r;-,.  FIG.  206.    A  bar  magnet 

If  a  magnet  is  dipped  into  iron  filings, 

the  filings  will  be  seen  to  cling  in  tufts  near  the  ends  but  scarcely 
at  all  near  the  middle  (Fig.  208).  These  places  near  the  ends  of 
a  magnet  at  which  its  strength  seems  to 
be  concentrated  are  called  the  poles  of  the 
magnet.  The  end  of  a  freely  swinging 
magnet  which  points  to  the  north  is  des- 
ignated as  the  north-seeking,  or  simply 
the  north  pole  (N) ;  and  the  other  end  as  the  south-seeking, 
or  the  south  pole  (S).  The  direction  in  which  a  compass  needle 
points  is  called  th 


FIG.  208.    Iron  filings  cling- 
ing to  bar  magnet 


FIG.  207.   A  horseshoe 
magnet 


270.  The  laws  of  magnetic  attrac- 
tion and  repulsion.  In  the  experiment 
with  the  iron  filings  no  particular 
difference  was  observed  between  the 

action  of  the  two  poles.    That  there  is  a  difference,  however, 
may  be  shown  by  experimenting  with  two  magnets,  either  of 
which  may  be  suspended  (see  Fig.  209). 
If  two  N  poles  are  brought  near  one  an- 
other, they  are  found  to  repel  each  other. 
The  S  poles  likewise  are  found  to  repel 
each  other.   But  the  ^Ypole  of  one  magnet 
is  found  to  be  attracted  by  the  S  pole 
of  another.  The  results  of  these  experi- 
ments may  be  summarized  in  a  general 
law :  Magnet  poles  of  like  kind  repel  each 
other,  while  poles  of  unlike  kind  attract. 
I    The  force  which  any  two  poles  exert 

/upon  each  other  has  been  found,  like  the  force  of  gravitation, 
to  vary  inversely  as  the  square  of  the  distance  between  them. 


FIG.  209.    Magnetic  at- 
tractions and  repulsions 


-=^    NT  ^T^  -/\.  v*  f       •-    r 
GENERAL  PROPERTIES  OF  MAGNETS  209 

A  unit  pole  is  defined  as  a  pole  which  when  placed  at  a 
distance  of  1  centimeter  from  an  exactly  similar  pole  repels  it 
with  a  force  of  1  dyne. 

271.  Magnetic    materials.     Iron    and    steel    are    the    only 
substances  which  exhibit  magnetic  properties  to  any  marked 
degree.    Nickel  and  cobalt  are  also  attracted  appreciably  by 
strong  magnets.    Bismiitlix_aiiti]iiQny,  and  a  number  of  other 
substances  are  actually  repelled  instead  of  attracted,  but  the 
effect  is  very  small.    It  has  recently  been  found  possible  to 
make  quite  strongly  magnetic  alloys  out  of  certain  nonmag- 
netic materials.   For  example,  a  mixture  of  65%  copper,  27% 
manganese,  and  8%   aluminum  is  quite  strongly  magnetic. 
These  are  called  Heusler  alloys.   For  practical  purposes,  how- 
ever, iron  and  steel  may  be  considered  as  the  only  magnetic 
materials. 

272.  Magnetic  induction.    If  a  small  unmagnetized  nail  is 
suspended  from  one  end  of  a  bar  magnet,  it  is  found  that 
a  second  nail  may  be  suspended  from  this  first  nail,  which 
itself  acts  like  a  magnet,  a  third  from  the 

second,  etc.,  as  shown  in  Fig.  210.    But  if 

the  bar  magnet  is  carefully  pulled  away 

from  the  first  nail,  the  others  will  instantly 

fall  away  from  each  other,  thus  showing' 

that  the  nails  were  strong  magnets  only 

so  long  as  they  were  in  contact  with  the 

bar  magnet.  ABMi^S^^aim.nay  be 

thus  magnetized  temporarily 'by  holding  it  in 

contact  with  a  permanent  magnet.    Indeed,  it  is  not  necessary 

that  there  be  actual  contact,  for  if  a  nail  is  simply  brought 

near  to  the  permanent  magnet  it  is  found  to  become  a  magnet. 

This  may  be  proved  by  presenting  some  iron  filings  to  one 

end  of  a  nail  held  near  a  magnet  in  the  manner  shown  in 

Fig.  211.    Even  inserting  a  plate  of  glass,  or  of  copper,  or 

of  any  other  material  except  iron  between  S  and  N  will  not 


210  MAGNETISM 

change  appreciably  the  number  of  filings  which  cling  to  the 
end  of  $',  a  fact' which  shows  that  nonmagnetic  materials  are 
transparent  to  magnetic  fexces.  But  as  soon  as  the  permanent 
magnet  Ts^rembved  most  of  the  filings  will  fall.  Magnetism 
I  produced  in  this  way  by  the  mere  presence  of  adjacent  magnets, 
with  or  without  contact,  is  called  induced  magnetism.  If  the 
induced  magnetism  of  the  nail  in  Fig.  211  is  tested  with  a 
compass  needle,  it  is  found  that  the  remote  induced  pole  is 
of  the  same  kind  as  the  inducing  pole,  while 
the  near  pole  is  of  unlike  kind.  This  is  the 
general  law  of  magnetic  induction. 

Magnetic  induction  explains  the  fact  that 
a  magnet  attracts  an  unmagnetized  piece  of 
iron,  for  it  first  magnetizes  it  by  induction, 
so  that  the  near  pole  is  unlike  the  inducing 
pole,  and  the  remote  pole  like  the  inducing  FlG-  211-  Masnet- 

i  J    xi  Vu  vi  i          ism  induced   Wlth' 

pole;  and  then,  since  the  two  unlike  poles          out  contact 
are  closer  together  than  the  like  poles,  the 
attraction  overbalances  the  repulsion  and  the  iron  is  drawn 
toward  the  magnet.    Magnetic  induction  also  explains  the 
formation  of  the  tufts  of  iron  filings  shown  in  Fig.  208,  each 
little  filing  becoming  a  temporary  magnet  such  that  the  end 
which  points  toward  the  inducing  pole  is  unlike  this  pole, 
and  the  end  which  points  away  from  it  is  like  this  pole.    The 
bushlike  appearance  is  due  to  the  repulsive  action  which  the 
outside  free  poles  exert  upon  each  other. 

273.  Retentivity  and  permeability.  A  piece  of  soft  iron 
will  very  easily  become  a  strong  temporary  magnet,  but  when 
removed  from  the  influence  of  the  magnet  it  loses  practically 
all  of  its  magnetism.  On  the  other  hand,  a  piece  of  steel  will 
not  be  so  strongly  magnetized  as  the  soft  iron,  but  it  will  re- 
tain a  much  larger  fraction  of  its  magnetism  after  it  is  removed 
from  the  influence  of  the  permanent  magnet.  This  power_pf 
resisting  either  magnetization  or  demagnetization  is  called 


GENERAL  PROPERTIES  OF  MAGNETS 


211 


FIG.  212.   A  line  of  force  set  up 
by  the  magnet  AB 


retentivity.  Thus  ste^l  has  a  much  greater  retentivity  than 
wrought  iron,  and,  in  general,  the  harder  the  steel  the  greater 
its  retentivity. 

A  _substaiice_jwhich  has  the  property  of  becoming  strongly 
magnetic  under  the  influence  of  a  permanent  magnet,  whether 
it  has  a  high  retentivity  or  not,  isj^id_io-possess  permeability  in 
large  degree.  Thus  iron  is  much  more  permeable  than  nickel. 

274.  Magnetic  lines  of  force.  If  we  could  separate  the  N 
and  S  poles  of  a  small  magnet  so  as  to  get  an  independent 
AT  pole,  and  were  to  plac&Nthis  N 
pole  near  the  N  pole  of  a  bar 
magnet,  it  would  move  over  to 
the  S  pole  along  some  curved 
path  similar  to  that  shown  in 
Fig.  212.  The  reason  it  would 
move  in  a  curved  path  is  that  it 
would  be  simultaneously  repelled  by  the  N  pole  of  the  bar 
magnet,  and  attracted  by  its  S  pole,  and  the  relative  strengths 
of  these  two  forces  would  continually  change,  as  the  relative 
distances  of  the  moving  pole  from  these  two  poles  changed. 

To  verify  this  conclusion  let  a  strongly  magnetized  sewing  needle  be 
floated  in  a  small  cork  in  a  shallow  dish  of  water,  and  let  a  bar  or 
horseshoe  magnet  be  placed  just  above  or  just  'beneath  the  dish  (see 
Fig.  213).  The  cork  and  needle  will  then  move  as  would  an  independent 
pole,  since  the  remote  pole  of  the 
needle  is  so  much  farther  from  the 
magnet  than  the  near  pole  that  its  in- 
fluence on  the  motion  is  very  small. 
The  cork  will  actually  be  found  to  FlG  21g  Showing  direction  of  a 
move  in  a  curved  path  from  ^V  to  5.  motion  of  an  isolated  pole  near 

Any- pathjs hiah_an  independ- 
ent N  pole  would  take  in  going  from  Nto  S  is  called  a  line  of 
force.    The  simplest  way  of  finding  the  direction  of  this  path 
at  any  point  near  a  magnet  is  to  hold  a  short  compass  needle 
at  the  point  considered.    The  compass  needle  sets  itself  along 


212 


MAGNETISM 


the  line  in  which  its  poles  would  move  if  independent,  that 
is,  along  the  line  of  force  which  passes  through  the  given 
point  (see  C,  Fig.  212). 

275.  Fields  of  force.  The  region  about  a  magnet  in  which 
its  magnetic  forces  can  be  detected  is  called  its  field  of  force. 
The  easiest  way  of  gaining  an  idea  of  the  way  in  which  the 
lines  of  force  are  arranged  in  the  magnetic  field  about  any 
magnet  is  to  sift  iron  filings  upon  a  piece  of  paper  placed 
immediately  over  the  magnet.  Each  little  filing  becomes  a 
temporary  magnet  by  induction,  and  therefore,  like  the  com- 
pass needle,  sets  itself  in  the  direction  of  the  line  of  force  at 


FIG.  214.    Arrangement  of  iron          FIG.  215.    Ideal  diagram  of  field 
filings  about  a  bar  magnet  of  a  bar  magnet 

the  point  where  it  is.  Fig.  214  shows  how  the  filings  arrange 
themselves  about  a  bar  magnet.  Fig.  215  is  the  correspond- 
ing ideal  diagram,  showing  the  lines  of  force  emerging  from 
the  N  pole  and  passing  about  in  curved  paths  to  the  JS  pole. 
It  is  customary  to  imagine  these  lines  as  returning  through 
the  magnet  from  S  to  N  in  the  manner  shown,  so  that  each 
line  is  thought  of  as  a  closed  curve.  This  convention  was 
introduced  by  Faraday,  and  has  been  found  of  great  assistance 
in  correlating  the  facts  of  magnetism. 

A  magnetic  field  of  unit  strength  is  defined  as  a  field  in  which 
a  unit  magnet  pole  experiences  1  dyne  of  force.  It  is  customary 
to  represent  graphically  such  a  field  by  drawing  one  line  per 


GENERAL  PROPERTIES  OF  MAGNETS 


213 


square  centimeter  through  a  surf  ace  such  as  ABCD  (Fig.  216) 
taken  at  right  angles  to  the  lines  of  force.  If  a  unit  N  pole 
between  ^and  S  (Fig.  216)  were  pushed  toward  S  with  a  force 
of  1000  dynes,  the  strength  of  the  field  would  be  1000  units 
and  it  would  be  represented  by 
1000  lines  per  square  centimeter. 
276.  Molecular  nature  of  mag- 
netism. If  a  small  test  tube  full 
of  iron  filings  be  stroked  from 
end  to  end  with  a  magnet,  it  will 
be  found  to  have  become  itself  a 
magnet ;  but  it  will  lose  its  mag-  FIG  216  The  strength  of  a  mag_ 
netism  as  soon  as  the  filings  are  netic  field  is  represented  by  the 
shaken  up.  If  a  magnetized  knit-  number  of  lines  of  force  per  square 

TT      .     i       ,     n         -IT          .,  centimeter 

ting  needle  is  heated  red-hot,  it 

will  be  found  to  have  lost  its  magnetism  completely.  Again,  if 
such  a  needle  is  jarred,  or  hammered,  or  twisted,  the  strength 
of  its  poles,  as  measured  by  their  ability  to  pick  up  tacks  or 
iron  filings,  will  be  found  to  be  greatly  diminished. 

These  facts  point  to  the  conclusion  that  magnetism  has 
something  to  do  with  the  arrangement  of  the  molecules,  since 
causes  which  violently  dis- 
turb the  molecules  of  a  mag- 
net weaken  its  magnetism. 
Again,  if  a  magnetized  needle 
is  broken,  each  part  will 
be  found  to  be  a  complete 

magnet;  that  is,  two  new  poles  will  appear  at  the  point  of 
breaking,  a  new  ^pole  on  the  part  which  has  the  original  S  pole, 
and  a  new  S  pole  on  the  part  which  has  the  original  N  pole. 
The  subdivision  may  be  continued  indefinitely,  but  always 
with  the  same  result,  as  indicated  in  Fig.  217.  This  suggests 
that  the  molecules  of  a  magnetized  bar  may  themselves  be  little 
magnets  arranged  in  rows  with  their  opposite  poles  in  contact. 


FIG.  217.   Effect  of  breaking  a  magnet 


I 


g  ^C^aa^^gnD^^m^^   g 


214  MAGNETISM 

If  an  unmagnetized  piece  of  hard  steel  is  pounded  vigorously 
while  it  lies  between  the  poles  of  a  magnet,  or  if  it  is  heated 
to  redness  and  then  allowed  to  cool  in  this  position,  it  will  be 
found  to  have  become  magnetized.  This  suggests  that  the 
molecules  of  the  steel  are  magnets  even  when  the  bar  as  a 
whole  is  not  magnetized,  and  that  magnetization  may  consist 
in  causing  them  to  arrange  themselves  in  rows,  end  to  end, 
just  as  the  magnetization  of  the  tube  of  iron  filings  mentioned 
above  was  due  to  a  special  arrangement  of  the  filings. 

277.  Theory  of  magnetism.  In  an  unmagnetized  bar  of  iron 
or  steel  it  is  probable  then  that  the  molecules  themselves  are 
tiny  magnets  which  are 
arranged  either  haphaz- 
ard, or  in  little  closed 
groups  or  chains,  as  in 
Fig.  218,  SO  that,  on  the  FlG-  218-  Arrangement  of  molecules  in  an 

unmagnetized  iron  bar 
whole,    opposite    poles 

neutralize  each  other  throughout  the  bar.  But  when  the  bar 
is  brought  near  a  magnet,  the  molecules  are  swung  around 
by  the  outside  magnetic  force  into  some  such  arrangement  as 
that  shown  in  Fig.  219,  in  which  the  opposite  poles  completely 
neutralize  each  other  only  in  the  middle  of  the  bar.  According 
to  this  view,  heating 
and  jarring  weaken  the 
magnet  because  they 
tend  to  shake  the  mole- 
cules out  of  alignment.  FIG.  219.  Arrangement  of  molecules  in  a 
On  the  Other  hand,  magnetized  iron  bar 

heating  and  jarring  facilitate  magnetization  when  the  bar  is 
between  the  poles  of  a  magnet  because  they  assist  the  mag- 
netizing force  in  breaking  up  the  molecular  groups  and  chains 
and  getting  the  molecules  into  alignment.  Soft  iron  has  higher 
permeability  than  hard  steel  because  the  molecules  of  the 
former  substance  are  much  easier  to  swing  into  alignment 


--  -  -  **»*w»**,rfv>*5K 


ODD  an  an  ara  am  en  cm  am  an  am  an  am  am  aft  an  am  nnm 


TERRESTRIAL  MAGNETISM  215 

than  those  of  the  latter  substance.  Steel  has  a  very  much 
greater  retentivity  than  soft  iron  because  its  molecules  are 
not  so  easily  moved  out  of  position  once  they  have  been 
aligned. 

278.  Saturation.    Strong  evidence   for  the  correctness  of 
the  above  view  is  found  in  the  fact  that  a  piece  of  iron  or 
steel  cannot  be  magnet- 
ized beyond  a  certain 

limit,  no  matter  how 
strong  is  the  magnetiz- 
ing force.  This  limit  FIG.  220.  Arrangement  of  molecules  in  a 

i     i  i  saturated  magnet 

probably    corresponds 

to  the  condition  in  which  the  axes  of  all  the  molecules  are 
brought  into  parallelism,  as  in  Fig.  220.  The  magnet  is  then 
said  to  be  saturated,  since  it  is  as  strong  as  it  is  possible 
to  make  it. 

TERRESTRIAL  MAGNETISM 

279.  The  earth's  magnetism.   The  fact  that  a  compass  needle 
always  points  norjbh  and  south,  or  approximately  so,  indicates 
that  the  earth  itself  is  a  great  magnet,  having  an  S  pole  near 
the  geographical  north  pole,  and  an  JVpole  near  the  geograph- 
ical south  pole;  for  the  magnetic  pole  of  the  earth  which  is 
near  the  geographical  north  pole  must  of  course  be  unlike  the 
pole  of  a  suspended  magnet  which  points  toward  it,  and  the  pole 
of  the  suspended  magnet  which  points  toward  the  north  is  the 
one  which  by  convention  it  has  been  decided  to  call  the  JVpole. 
The  magnetic  pole  of  the  earth  which  is  near  the  north  geo- 
graphical pole  was  found  in  1831   by  Sir  James  Ross  in 
Boothia  Felix,  Canada,  latitude  70°  30'  N.,  longitude  95°  W. 
It  was  located  again  in  1905  by  Captain  Amundsen  (the  dis- 
coverer of  the  geographical  south  pole,  1912)  at  a  point  a 
little  farther  west.    Its  approximate  location  is  70°  5'  N.  and 
96°  46'  W.    It  is  probable  that  it  shifts  its  position  slowly. 


216  MAGNETISM 

280.  Declination.    The  earliest  users  of  the  compass  were 
aware  that  it  did  not  point  exactly  north ;  but  it  was  Columbus 
who,  on  his  first  voyage  to  America,  made  the  discovery,  much 
to  the  alarm  of  his  sailors,  that  the  direction  of  the  compass 
needle  changes  as  one  moves  about  over  the  earth's  surface. 
The  chief  reason  for  this  variation  is  found  in  the  fact  that  the 
magnetic  poles  do  not  coincide  with  the  geographical  poles ;  but 
there  are  also  other  causes,  such  as  the  existence  of  large  de- 
posits of  iron  ore,  which  produce  local  effects  upon  the  needle. 
The  number  of  degrees  by  which,  at  a  given  point  on  the  earth, 
the  needle  varies  from  a  true  north-and-south  line  is  called  its 
declination  at  that  point.    Lines  drawn  over  the  earth  through 
points  of  equal  declination  are  called  isogonic  lines. 

281.  Dip  Of  the  compass  needle.    Let  an  unmagnetized'  knitting 
needle  a  (Fig.  221)  be  thrust  through  a  cork,  and  let  a  second  needle 
b  be  passed  through  the  cork  at  right  angles  to  a,  and  as  close  to  it 
as  possible.    Let  a  pin  c  be  adjusted  until  the 

system  is  in  neutral  equilibrium  about  b  as  an 

axis,  when  a  is  pointing  east  and  west.    Then 

let  a  be  carefully  magnetized  by  stroking  one 

end  of  it  from  the  middle  out  with  the  N  pole 

of  a  strong  magnet,  and  the  other  end  from  ^   Arrangement 

the  middle  out  with  the  S  pole  of  the  same  for  silowmg  <jip 

magnet.    When  now  the  needle  is  replaced  on 

its  supports  and  turned  into  a  north-and-south  position,  its  N  pole  will 

be  found  to  dip  so  as  to  cause  the  needle  to  make  an  angle  of  60°  or 

70°  with  the  horizontal. 

The  experiment  shows  that  in  this  latitude  the  earth's  mag- 
netic lines  make  a  large  angle  with  the  horizontal.  This  angle 
between  the  earth's  surface  and  the  direction  of  the  magnetic 
lines  is  called  the  dip,  or  inclination,  of  the  needle.  At  Wash- 
ington it  is  71°  5'  and  at  Chicago  72°  50'.  At  the  magnetic 
pole  it  is  of  course  90°,  and  at  the  so-called  magnetic  equator, 
which  is  an  irregular  curved  line  near  the  geographical  equator, 
the  dip  is  0°. 


TERRESTRIAL  MAGNETISM  217 

282.  The  earth's  inductive  action.  That  the  earth  acts  like  a 
great  magnet  may  be  very  strikingly  shown  in  the  following  way : 

Let  a  steel  rod,  for  example  a  tripod  rod,  be  held  parallel  to  the 
earth's  magnetic  lines  (the  north  end  slanting  down  at  an  angle  of  about 
70°  or  75°)  and  struck  a  few  sharp  blows  with  a  hammer.  The  rod  will 
be  found  to  have  become  a  magnet  with  its  upper  end  an  S  pole,  like 
the  north  pole  of  the  earth,  and  its  lower  end/au  N  pole.  If  the  rod  is 
reversed  and  tapped  again  with  the  hammer,  its  magnetism  will  be  re- 
versed. If  held  in  an  east-and-west  position  and  tapped,  it  will  become 
demagnetized,  as  will  be  shown  by  the  fact  that  either  end  of  it  will 
attract  either  end  of  a  compass  needle. 

QUESTIONS  AND  PROBLEMS 

1.  If  a  bar  magnet  is  floated  on  a  piece  of  cork,  will  it  tend  to  float 
toward  the  north ?    Why? 

2.  Will  a  bar  magnet  pull  a  floating  compass  needle  toward  it? 
Compare  the  answer  to  this  question  with  that  to  the  preceding  one. 

3.  Why  should  the  needle  used  in  the  experiment  of  §  281  be  placed 
east  and  west,  when  adjusting  for  neutral  equilibrium,  before  it  is 
magnetized  ? 

4.  The  dipping  needle  is  suspended  from  one  arm  of  a  steel-free 
balance  and  carefully  weighed.   It  is  then  magnetized.   Will  its  apparent 
weight  increase  ? 

5.  Explain,  on  the  basis  of  induced  magnetization,  the  process  by 
which  a  magnet  attracts  a  piece  of  soft  iron.        # 

6.  When  a  piece  of  soft  iron  is  made  a  temporary  magnet  by  bring- 
ing it  near  the  N  pole  of  a  bar  magnet,  will  the  end  of  the  iron  nearest 
the  magnet  be  an  N  or  an  S  pole  ? 

7.  Devise  an  experiment  which  will  show  that  a  piece  of  iron  attracts 
a  magnet  just  as  truly  as  the  magnet  attracts  the  iron. 

8.  How  would  an  ordinary  compass  needle  act  if  placed  over  one  of 
the  earth's  magnetic  poles?   How  would  a  dipping  needle  act  at  these 
points  ? 

9.  Do  the  facts  of  induction  suggest  to  you  any  reason  why  a  horse- 
shoe magnet  retains  its  magnetism  better  when  a  bar  of  soft  iron  (a 
keeper,  or  armature)  is  placed  across  its  poles  than  when  it  is  not  so 
treated?    (See  Fig.  219.) 

10.  With  what  force  will  an  N  magnetic  pole  of  strength  6  attract, 
at  a  distance  of  5  cm.,  an  S  pole  of  strength  1  ?  of  strength  9  ? 


CHAPTER  XII 

STATIC  ELECTRICITY 

GENERAL  FACTS  OF  ELECTRIFICATION 

283.  Electrification  by  friction.  If  a  piece  of  hard  rubber  or 
a  stick  of  sealing  wax  is  rubbed  with  flannel  or  cat's  fur  and 
then  brought  near  some  dry  pith  balls,  bits  of  paper,  or  other 
light  bodies,  these  bodies  are  found  to  jump  toward  the  rod. 
This  sort  of  attraction,  so  familiar  to  us  from  the  behavior  of 
our  hair  in  winter  when  we  comb  it  with  a  rubber  comb,  was 
observed  as  early  as  600  B.C.,  when  Thales  of  Greece  com- 
mented upon  the  fact  that  rubbed  amber  draws  to  itself  threads 
and  other  light  objects.  It  was  not,  however,  until  1600  A.D. 
that  Dr.  William  Gilbert,  physician  to  Queen  Elizabeth,  and 
sometimes  called  the  father  of  the  modern  science  of  electricity 
and  magnetism,  discovered  that  the  effect  could  be  produced 
by  rubbing  together  a  great  variety  of  other  substances  besides 
amber  and  silk,  such,  for  example,  as  glass  and  silk,  sealing 
wax  and  flannel,  hard  rubber  and  cat's  fur,  etc. 

Gilbert  named  the  effect  which  was  produced  upon  these 
various  substances  by  friction,  electrification,  after  the  Greek 
name  electron,  meaning  "  amber."  Thus  a  body  which,  like 
rubbed  amber,  has  been  endowed  with  the  property  of  attracting 
light  bodies  is  said  to  have  been  electrified,  or  to  have  been  given 
a  charge  of  electricity.  In  this  statement  nothing  whatever  is 
said  about  the  nature  of  electricity.  We  simply  define  an 
electrically  charged  body  as  one  which  has  been  put  into  the 
condition  in  which  it  acts  toward  light  bodies  like  the  rubbed 
amber  or  the  rubbed  sealing  wax.  To  this  day  we  do  not  know 

218 


WILLIAM  GILBERT  (1540-1603) 

English  physician  and  physicist;  first  Englishman  to  appreciate 
fully  the  value  of  experimental  observations;  first  to  discover 
through  careful  experimentation  that  the  compass  points  to  the 
north,  not  because  of  some  influence  of  the  stars,  but  because  the 
earth  is  itself  a  great  magnet ;  first  to  use  the  word  "  electricity  "  ; 
first  to  discover  that  electrification  can  be  produced  by  rub- 
bing a  great  many  different  kinds  of  substances ;  author  of  the 
epoch-making  book  entitled  "De  Magnete,  etc.,"  published  in 
London  in  1600 


GENERAL  FACTS  OF  ELECTRIFICATION        219 

with  certainty  what  the  nature  of  electricity  is,  but  we  are  fairly 
familiar  with  the  laws  which  govern  its  action.  It  is  to  these 
laws  that  attention  will  be  mainly  devoted  in  the  following 
sections. 

284.  Positive  and  negative  electricity.   Let  a  pith  ball  suspended 

by  a  silk  thread,  as  in  Fig.  222,  be  touched  to  a  glass  rod  which  has  been 
rubbed  with  silk  and  thus  been  put  into  the  condition  in  which  it  is 
strongly  repelled  by  this  rod.  Next 
let  a  stick  of  sealing  wax  or  an  ebon- 
ite rod  which  has  been  rubbed  with 
cat's  fur  or  flannel  be  brought  near 
the  charged  ball.    It  will  be  found 
that  it  is  not  repelled,  but,  on  the 
contrary,  is  very  strongly  attracted. 
Similarly,  if  the  pith  ball  has  touched 
the  sealing  wax  so  that  it  is  repelled       FlG  222    Pith.ban  electroscope 
by  it,  it  is  found  to  be   strongly 

attracted  by  the  glass  rod.  Again,  two  pith  balls  both  of  which  have 
been  in  contact  with  the  glass  rod  are  found  to  repel  each  other,  while 
pith  balls  one  of  which  has  been  in  contact  with  the  glass  rod  and  the 
other  with  the  sealing  wax  attract  each  other. 

Evidently,  then,  the  electrifications  which  are  imparted  to 
glass  by  rubbing  it  with  silk  and  to  sealing  wax  by  rubbing 
it  with  flannel  are  opposite  in  the  sense  that  an  electrified 
body  that  is  attracted  by  one  is  repelled' by  the  other.  We 
say,  therefore,  that  there  are  two  kinds  of  electrification,  and 
we  arbitrarily  call  one  positive  and  the  other  negative.  Thus  a 
positively  electrified  body  is  one  which  acts  with  respect  to  other 
electrified  bodies  like  a  glass  rod  which  has  been  rubbed  with 
silk,  and  a  negatively  electrified  body  is  one  which  acts  like  a 
piece  of  sealing  wax  which  has  been  rubbed  with  flannel.  These 
facts  and  definitions  may  then  be  stated  in  the  following  gen- 
eral law :  Electrical  charges  of  like  kind  repel  each  other,  while 
charges  of  unlike  kind  attract  each  other.  The  forces  of  attrac- 
tion or  repulsion  are  found,  like  those  of  gravitation  and 
magnetism,  to  decrease  as  the  square  of  the  distance  increases. 


220  STATIC  ELECTRICITY 

285.  Measurement  of  electrical  quantities.    The  fact  of  attraction  and 
repulsion  is  taken  as  the  basis  for  the  definition  and  measurement  of 
so-called  quantities  of  electricity.    Thus  a  small  charged  body  is  said  to 
contain  1  unit  of  electricity  when  it  will  repel  an  exactly  equal  and 
similar  charge  placed  1  centimeter  away  with  a  force  of  1  dyne.    The 
p.umber  of  units  of  electricity  on  any  charged  body  is  then  measured 
by  the  force  which  it  exerts  upon  a  unit  charge  placed  at  a  given  distance 
from  it;  for  example,  a  charge  which  at  a  distance  of  10  centimeters 
repels  a  unit  charge  with  a  force  of  1  dyne  contains  100  units  of  elec- 
tricity, for  this  means  that  at  a  distance  of  1  centimeter  it  would  repel 
the  unit  charge  with  a  force  of  100  dynes  (see  §  284). 

286.  Conductors   and  nonconductors.    Let  an   electroscope  E 

(Fig.  223),  consisting  of  a  pair  of  gold  leaves  a  and  b,  suspended  from 

an  insulated  metal  rod  r,  and  protected  from  air  currents  by  a  case  J, 

be  connected  with  the  metal  ball  B  by  means  of  a  wire.   Let  an  ebonite 

rod  be  now  electrified  and  rubbed 

over  B.    The  immediate  divergence 

of  the  gold  leaves  will  show  that  a 

portion  of  the  electric  charge  placed 

upon  B  has  been  carried  by  the  wire 

to  the  gold  leaves,  where  it  causes 

them  to  diverge  in  accordance  with 

the  law  that   bodies   charged  with 

the  same   kind   of  electricity  repel 

each  other. 

Let  the  experiment  be  repeated  FIG.  223.  Illustrating  conduction 
when  E  and  B  are  connected  with  a 

thread  of  silk  or  a  long  rod  of  wood  instead  of  the  metal  wire.  No 
divergence  of  the  leaves  will  be  observed.  If  a  moistened  thread  con- 
nects E  and  B,  the  leaves  will  be  seen  to  diverge  slowly  when  the  ball  B 
is  charged,  showing  that  a  charge  is  carried  slowly  by  the  moist  thread. 

These  experiments  make  it  clear  that  while  electric  charges 
pass  with  perfect  readiness  from  one  point  to  another  in  a  wire, 
they  are  quite  unable  to  pass  along  dry  silk  or  wood,  and  pass 
with  difficulty  along  moist  silk.  We  are  therefore  accustomed 
to  divide  substances  into  two  classes,  conductors  and  noncon- 
ductors, or  insulators,  according  to  their  ability  to  transmit  elec- 
trical charges  from  point  to  point.  Thus  metals  and  solutions 


GENEBAL  FACTS  OF  ELECTRIFICATION        221 

of  salts  and  acids  in  water  are  all  conductors  of  electricity, 
while  glass,  porcelain,  rubber,  mica,  shellac,  wood,  silk,  vase- 
line, turpentine,  paraffin,  and  oils  generally  are  insulators. 
No  hard-and-fast  line,  however,  can  be  drawn  between  con- 
ductors and  nonconductors,  since  all  so-called  insulators 
conduct  to  some  slight  extent,  while  the  so-called  conductors 
differ  greatly  among  themselves  in  the  facility  with  which 
they  transmit  charges. 

The  fact  of  conduction  brings  out  sharply  one  of  the  most 
essential  distinctions  between  electricity  and  magnetism.  Mag- 
netic poles  exist  only  in  iron  and  steel,  while  electrical  charges 
may  be  communicated  to  any  body  whatever,  provided  it  is 
insulated.  These  charges  pass  over  conductors,  and  can  be 
transferred  by  contact  from  one  body  to  any  other,  while 
magnetic  poles  remain  fixedjn^  position,  and  are  wholly  unin- 
fluenced by  contact  with  other  bodies,  unless  these  bodies 
themselves  are  magnets. 

287.  Electrostatic  induction.  Let  the  ebonite  rod  be  electrified  by 
friction  and  slowly  brought  toward  the  knob  of  the  gold-leaf  electroscope 
(Fig.  224).  The  leaves  will  be  seen 
to  diverge,  even  though  the  rod  does 
not  approach  to  within  a  foot  of  the 
electroscope. 

This  makes  it  clear  that  the 
mere  influence  which  an  elec- 
tric charge  exerts  upon  a  con- 
ductor placed  in  its  neighborhood 
is  able  to  produce  electrification  Fjo_  ^  Illustrating  in(iuotioa 
in  that  conductor.  This  method 
of  producing  electrification  i^  called  electrostatic  induction. 

As  soon  as  the  charged  rod  is  removed  the  leaves  will  be 
seen  to  collapse  completely.  This  shows  that  this  form  of  elec- 
trification is  only  a  temporary  phenomenon  which  is  due  simply 
to  the  presence  of  the  charged  body  in  the  neighborhood. 


222  STATIC  ELECTRICITY 

288.  Nature  of  electrification  produced  by  induction.    Let  a 

metal  ball  A  (Fig.  225)  be  strongly  charged  by  rubbing  it  with  a  charged 
rod,  and  let  it  then  be  brought  near       A  aA  ?  ?s? 

an  insulated*  metal  body  B,  which  is      /^~~\    & ^ S* 

provided  with  pith  balls  or  strips  of      V        J    ^— 

paper  a,  b.  c.  as  shown.   The  divergence 

*  ,         .,,    ,         .-,        ,,  -,      4.        FIG.  225.    Nature  of  induced 

of  a  and  c  will  show  that  the  ends  of 

B  have  received  electrical  charges  be- 
cause of  the  presence  of  A,  while  the  failure  of  b  to  diverge  will  show 
that  the  middle  of  B  is  uncharged.    Further,  the  rod  which  charged  A 
will  be  found  to  repel  c,  but  to  attract  a. 

We  conclude,  therefore,  that  when  a  conductor  is  brought 
near  a  charged  body,  the  end  away  from  the  inducing  charge 
is  electrified  with  the  same  kind  of  electricity  as  that  on  the  in- 
ducing body,  while  the  end  toward  the  inducing  body  receives 
electricity  of  opposite  kind. 

289.  Two-fluid  theory  of  electricity.    We  can  describe  the 
facts  of  induction  conveniently  by  assuming  that  in  every  con- 
ductor there  exists  an  equal  number  of  positively  and  nega- 
tively charged  corpuscles  which  are  very  much  smaller  than 
atoms,  and  which  are  able  to  move  about  freely  within  the 
conductor.   When  no  electrified  body  is  near  the  conductor  B, 
it  appears  to  have  no  charge  at  all,  because  all  the  little  posi- 
tive charges  within   it  counteract  the   effects  upon  outside 
bodies  of  all  the  little  negative  charges.    But  as  soon  as  an 
electrical  charge  is  brought  near  B,  it  drives  as  far  away  as 
possible  the  corpuscles  which  carry  charges  of  sign  like  its 
own,  while  it  attracts  the  corpuscles  of  unlike  sign.    B  there- 
fore becomes  electrified  like  A  at  its  remote  end  and  unlike 
A  at  its  near  end.    As  soon  as  the  inducing  charge  is  removed, 
B  immediately  becomes  neutral  again  because  the  little  posi- 
tive and  negative  corpuscles  come  together  under  the  influence 

*  Sulphur  is  practically  a  perfect  insulator  in  all  weathers,  wet  or  dry.  Metal 
conductors  of  almost  any  shape  resting  upon  pieces  of  sulphur  will  serve  the 
purposes  of  this  experiment  in  summer  or  winter. 


GENERAL  FACTS  OF  ELECTRIFICATION        223 

of  their  mutual  attractions.  This  picture  of  the  mechanism  of 
electrification  by  induction  is  a  modern  modification  of  the 
so-called  two-fluid  theory  of  electricity,  which  conceived  of  all 
conductors  as  containing  equal  amounts  of  two  weightless 
electrical  fluids  called  positive  electricity  and  negative  elec- 
tricity. Although  it  is  now  quite  improbable  that  this  theory 
represents  the  actual  conditions  within  a  conductor,  yet  we 
are  able  to  say  with  perfect  positiveness  that  the  electrical  be- 
havior of  a  conductor  is  exactly  what  it  would  be  if  it  did  contain 
equal  amounts  of  positive  and  negative  electrical  fluids,  or  equal 
numbers  of  minute  positive  and  negative  corpuscles  which  are 
free  to  move  through  the  conductor  under  the  influence  of 
outside  electrical  forces.  Furthermore,  since  the  real  nature 
of  electricity  has  been  altogether  unknown,  it  has  gradually 
become  a  universally  recognized  convention  to  speak  of  the 
positive  electricity  within  a  conductor  as  being  repelled  to  the 
remote  end,  and  the  negative  electricity  as  attracted  to  the  near 
end  by  an  outside  positive  charge,  and  vice  versa.  This  does 
not  imply  acceptance  of  the  two-fluid  theory.  It  is  merely  a 
way  of  describing  the  fact  that  the  remote  end  does  acquire 
a  charge  like  that  of  the  inducing  body,  and  the  near  end  a 
charge  unlike  that  of  the  inducing  body. 

290.  The  electron  theory.  A  slightly  different  theory,  called 
the  one-fluid  theory,  was  -originally  suggested  by  Benjamin 
Franklin,  and  in  the  following  modified  form  is  now  pretty  gen- 
erally held  by  physicists.  The  atoms  of  all  substances  are  now 
known  to  contain  as  constituents  both  positive  and  negative 
electricity,  the  latter  existing  in  the  form  of  minute  corpuscles 
or  electrons,  each  of  which  has  a  mass  of  about  1  ^  Q  that  of 
the  hydrogen  atom.  These  electrons  are  probably  grouped  in 
some  way  about  the  positive  electricity  as  a  nucleus.  The 
sum  of  the  negative  charges  of  these  electrons  is  supposed  to 
be  just  equal  to  the  positive  charge  of  the  nucleus,  so  that  in 
its  normal  condition  the  whole  atom  is  neutral  or  uncharged. 


224  STATIC  ELECTRICITY 

But  in  conductors  electrons  are  continually  getting  loose  from 
the  atoms  and  reentering  other  atoms,  so  that  at  any  given 
instant  there  are  always  in  every  conductor  a  number  of  free 
negative  electrons  and  a  corresponding  number  of  atoms  which 
have  lost  electrons  and  which  are  therefore  positively  charged. 
Such  a  conductor  would,  as  a  whole,  show  no  charge  of  either 
positive  or  negative  electricity.  But  as  soon  as  a  body  charged, 
for  example  negatively,  is  brought  near  such  a  conductor,  the 
negatively  charged  electrons  stream  away  to  the  remote  end, 
leaving  behind  them  the  positively  charged  atoms,  which  are 
not  free  to  move  from  their  positions.  On  the  other  hand,  if  a 
positively  charged  body  is  brought  near  the  conductor,  the 
negative  electrons  are  attracted  and  the  remote  end  is  left 
with  the  immovable  plus  atoms. 

The  only  advantage  of  this  theory  over  that  suggested  in 
§  289,  in  which  the  existence  of  both  positive  and  negative 
corpuscles  was  assumed,  is  that  there  is  much  direct  experi- 
mental evidence  for  the  existence  of  such  negatively  charged 
corpuscles  or  electrons  of  about  1  ^  0  the  mass  of  the  hydro- 
gen atom  (see  Chapter  XXI),  but  no  direct  evidence  as  yet  for 
the  existence  of  positively  charged  bodies  smaller  than  atoms. 

The  charge  of  one  electron  is  called  the  elementary  electri- 
cal charge.  Its  value  has  recently  (1913)  been  accurately 
measured.  There  are  2.095  billion  of  them  in  one  of  the 
units  denned  in  §  285.  Every  electrical  charge  consists  of  an 
exact  number  of  these  ultimate  electrical  atoms  scattered  over  the 
surface  of  the  charged  body. 

291.  Charging  by  induction.  Let  two  metal  balls  or  two  eggshells 
A  and  B,  which  have  been  gilded  or  covered  with  tin  foil,  be  suspended 
by  silk  threads  and  touched  together,  as  in  Fig.  226.  Let  a  positively 
charged  body  C  be  brought  near  them.  As  described  above,  A  and  B 
will  at  once  exhibit  evidences  of  electrification ;  that  is,  A  will  repel  a 
positively  charged  pith  ball,  while  B  will  attract  it.  If  C  is  removed 
while  A  and  B  are  still  in  contact,  the  separated  charges  reunite  and  A 


; 


BENJAMIN  FRANKLIN  (1706-1790) 

Celebrated  American  statesman,  philosopher,  auti.  scientist ;  born 
at  Boston,  the  sixteenth  child  of  poor  parents ;  printer  and  pub- 
lisher by  occupation;  pursued  scientific  studies  in  electricity  as 
a  diversion  rather  than  as  a  profession ;  first  proved  that  the  two 
coats  of  a  Ley  den  jar  are  oppositely  charged;  introduced  the 
terms  positive  and  negative  electricity ;  proved  the  identity  of 
lightning  and  frictional  electricity  by  flying  a  kite  in  a  thunder- 
storm and  drawing  sparks  from  the  insulated  lower  end  of  the 
kite  string ;  invented  the  lightning  rod ;  originated  the  one-fluid 
theory  of  electricity  which  regarded  a  positive  charge  as  indi- 
cating an  excess,  a  negative  charge  a  deficiency,  in  a  certain 
normal  amount  of  an  all-pervading  electrical  fluid 


GENERAL  FACTS  OF  ELECTRIFICATION        225 


and  B  cease  to  exhibit  electrification.  But  if  A  and  B  are  separated 
from  each  other  while  C  is  in  place,  A  will  be  found  to  be  permanently 
positively  charged  and  B  negatively  charged.  This  may  be  proved  either 

by  the  attractions  and  repulsions  which  ^ ( ^ 

they  show  for  charged  rods  brought  near 
them,  or  by  the  effects  which  they  pro- 
duce upon  a  charged  electroscope  brought 
into  their  vicinity,  the  leaves  of  the  latter 
falling  together  when  it  is  brought  near 
one  and  spreading  farther  apart  when 
brought  near  the  other. 


FIG.  226.    Obtaining  a  plus 
and  a   minus  charge  by  in- 
duction 


We  see/therefore^  that  if  we  cut 
in  two,  or  separate  into  two  parts,  <x 
conductor  while  it  is  under  the  influence  of  an  electric  charge, 
we  obtain  two  permanently  charged  bodies,  the  remoter  part 
having  a  charge  of  the  same  sign  as  that  of  the  inducing 
charge,  and  the  near  part  having  a  charge  of  unlike  sign. 

Let  the  conductor  B  (Fig.  227)  be  touched  by  the  finger  while  a 
charged  rod  C  is  near  it.  Then  let  the  finger  be  removed  and  after  it 
the  rod  C.  If  now  a  negatively  charged  pith  ball  is  brought  near  B,  it 
will  be  repelled,  showing  that  B  has 
become  negatively  charged.  In  this 
experiment  the  body  of  the  experi- 
menter corresponds  to  the  egg  A 
of  the  preceding  experiment,  and 
removing  the  finger  from  B  corre- 
sponds to  separating  the  two  egg- 
shells. Let  the  last  experiment  be 
repeated  with  only  this  modification,  that  B  is  touched  at  lj  rather  than 
at  a.  When  B  is  again  tested  with  the  pith  ball  it  will  still  be  found 
to  have  a  negative  charge,  exactly  as  when  the  finger  was  touched  at  a. 

We  conclude,  therefore,  that  no  matter  where  the  body  B  is 
touched,  the  sign  of  the  charge  left  upon  it  is  always  opposite  to 
that  of  the  inducing  charge.  This  is  because  the  negative  elec- 
tricity, that  is,  the  electrons,  can  under  no  circumstances  escape 
from  b  so  long  as  C  is  present,  for  they  are  "  bound  "  by  the 
attraction  of  the  positive  charge  on  C.  Indeed,  the  final 


FIG.  227.   A  body  charged  by  induc- 
tion has  a  charge  of  sign  opposite  to 
that  of  the  inducing  charge 


226  STATIC  ELECTRICITY 

negative  charge  on  B  is  due  merely  to  the  fact  that  the  positive 
charge  on  C  pulls  electrons  into  B  from  the  finger,  no  matter 
where  B  is  touched.  In  the  same  way* if  C  had  been  negative 
it  would  have  pushed  electrons  off  from  B  through  the  finger 
and  have  thus  left  B  positively  charged. 

,  292.  Charging  the  electroscope  by  induction.  Let  an  ebonite 
rod  which  has  been  rubbed  with  cat's  skin  be  brought  near  the  knob  of 
the  electroscope  (Fig.  224).  The  leaves  at  once  diverge.  Let  the  knob 
be  touched  with  the  finger  while  the  rod  is  held  in  place.  The  leaves 
will  fall  together.  Let  the  finger  be  removed  and  then  the  rod.  The 
leaves  will  fly  apart  again. 

* 
The  electroscope  has  been  charged  by  induction,  and  since 

the  charge  on  the  ebonite  rod  was  negative,  the  charge  on  the 
electroscope  must  be  positive.  If  this  conclusion  is  tested  by 
bringing  the  ebonite  rod  near  the  electroscope,  the  leaves  will 
fall  together  as  the  rod  approaches  the  knob.  How  does  this 
prove  that  the  charge  on  the  electroscope  is  positive  ? 

293.  Plus  and  minus  electricities  always  appear  simultane- 
ously and  in  equal  amounts.  Let  an  ebonite  rod  be  completely  dis- 
charged by  passing  it  quickly  through  a  Bunsen  flame.  Let  a  flannel 
cap  having  a  silk  thread  attached  be  slipped  over 
the  rod,  as  in  Fig.  228,  and  twisted  rapidly  around 
a  number  of  times.  When  rod  and  cap  together 
are  held  near  a  charged  electroscope,  no  effect  will 
be  observed ;  but  if  the  cap  is  pulled  off,  it  will 
be  found  to  be  positively  charged,  while  the  rod 
will  be  found  to  have  a  negative  charge.  FIG.  228.  Plus  and 

0.          .-,  ,,  ™  minus   electricities 

Since  the  two  together  produce  no  effect,    always    developed 

the  experiment  shows  that  the  plus  and  minus  in  equal  amounts 
charges  were  equal  in  amount.  This  experi- 
ment confirms  the  view  already  brought  forward  in  connection 
with  induction,  that  electrification  always  consists  in  a  separation 
of  plus  and  minus  charges  which  already  exist  in  equal  amounts 
within  the  bodies  in  which  the  electrification  is  developed. 


DISTRIBUTION  OF  CHARGE  227 

QUESTIONS  AND  PROBLEMS 

1.  Charge  a  gold-leaf  electroscope  by  induction  from  a  glass  rod. 
Warm  a  piece  of  paper  and  stroke  it  on  the  clothing.    Hold  it  over  the 
charged  electroscope.    If  the  divergence  of  the  gold  leaves  is  increased, 
is  the  charge  on  the  paper  +  or  —  ?   If  the  divergence  of  the  gold  leaves 
is  decreased,  what  is  the  sign  of  the  charge  on  the  paper  ? 

2.  Given  a  gold-leaf  electroscope,  a  glass  rod,  and  a  piece  of  silk, 
how,  in  general,  would  you  proceed  to  test  the  sign  of  the  electrification 
of  an  unknown  charge  ? 

3.  If  pith  balls,  or  any  light  figures,  are  placed  between  two  plates 
(Fig.  229),  one  of  which  is  connected  to  earth  and  the  other  to  one  knob 
of  an  electrical  machine  in  operation,  the  figures  will  bound  back  and 
forth  between  the  two  plates  as  long  as  the  machine 

is  operated.    Explain. 

4.  If  you  are  given  a  positively  charged  insulated 
sphere,  how  could  you  charge  two  other  spheres,  one 
positively  and  the  other  negatively,  without  diminish- 
ing the  charge  on  the  first  sphere  ? 

5.  If  you  bring  a  positively  charged  glass  rod  near 
the  knob  of  an  electroscope  and  then  touch  the  knob, 

why  do  you  not  remove  the  negative  electricity  which  j?1G  229 

is  on  the  knob? 

6.  In  charging  an  electroscope   by  induction,  why  must  the  finger 
be  removed  before  the  removal  of  the  charged  body  ?  , 

7.  If  you  hold  a  brass  rod  in  the  hand  and  rub  it  with  silk,  the  rod 
will  show  no  sign  of  electrification ;  but  if  you  hold  the  brass  rod  with 
a  piece  of  sheet  rubber  and  then  rub  it  with  silkj  you  will  find  it  elec- 
trified.   Explain. 

8.  Why  is  repulsion  between  an  unknown  body  and  an  electrified  pith 
ball  a  surer  sign  that  the  unknown  body  is  electrified  than  is  attraction  ? 

9.  State  as  many  differences  as  you  can  between  the  phenomena  of 
magnetism  and  those  of  electricity. 

DISTRIBUTION  OF  ELECTRIC  CHARGE  UPON  CONDUCTORS 
294.  Electric  charges  reside  only  upon  the  outside  surface  of 

conductors.  Let  a  deep  tin  cup  (Fig.  230)  be  placed  upon  an  insulating- 
stand  and  charged  as  strongly  as 'possible  either  from  an  ebonite  rod 
or  from  an  electrical  machine.  If  now  a  smooth  metal  ball  suspended  by  a 
silk  thread  is  touched  to  the  outside  of  the  charged  cup,  and  then  brought 
near  the  knob  of  a  charged  electroscope,  it  will  show  a  strong  charge ; 
but  if  it  is  touched  to  the  inside  of  the  cup,  it  will  show  no  charge  at  all. 


228  STATIC  ELECTKICITY 

These  experiments  show  that  an  electric  charge  resides 
entirely  on  the  outside  surface  vf  a  conductor.  This  is  a  result 
which  might  have  been  inferred  from 
the  fact  that  all  the  little  electrical 
charges  of  which  the  total  charge  is 
made  up  repel  each  other  and  there- 
fore move  through  the  conductor  un- 
til they  are,  on  the  average,  as  far 
apart  as  possible. 

295.  Density    of    charge    greatest    FlG-2so.  Proof  that  charge 

.         .  resides  on  surface 

where  curvature  of  surface  is  greatest. 

Since  all  of  the  parts  of  an  electric  charge  tend,  because  of 
their  mutual  repulsions,  to  get  as  far  apart  as  possible,  we 
should  infer  that  if  a  charge  of  either  sign  is  placed  upon  an 
oblong  conductor  like  that  of  Fig.  231,  (1),  it  will  distribute 
itself  so  that  the  electrification  at  the  ends  will  be  stronger 
than  that  at  the  middle. 

To  test  this  inference  let  a  proof  plane  (a  flat  metal  disk,  for  example  a 
cent,  provided  with  an  insulating  handle)  be  touched  to  one  end  of  such 
a  charged  body,  the  charge  conveyed  to  a  gold-  /j\ 

leaf  electroscope,  and  the  amount  of  separation       /'"~" "ZN 

of  the  leaves  noted.    Then  let  the  experiment      !A_ )} 

be  repeated  when  the  proof  plane  touches  the 

middle  of  the  body.     The  separation  of  the 

leaves  in  the  latter  case  will  be  found  to  be  very 

much  less  than  in  the  former.    If  we  should 

test  the  distribution  on  a  pear-shaped  body     ^^    retribution  of 

[Fig.  231,  (2)]   in  the  same  way,  we  should.    chargeoverobiongbodies 

find  the  density  of  electrification  considerably 

greater  on  the  small  end  than  on  the  large  one.    By  density  of  electrifi- 

.cation  is  meant  the  quantity  of  electricity  on  unit  area  of  the  surface. 

296.  Discharging  effect  of  points.    The  above  experiments 
indicate  that  if  one  end  of  a  pear-shaped  body  is  made  more 
and  more  pointed,  then  when  the  body  is  charged  the  elec- 
tric  density  on  this  end  will  become  greater  and  greater. 


DISTRIBUTION  OF  CHARGE  229 

The  following  experiment  will  show  what  happens  when  the 
conductor  is  provided  with  a  sharp  point. 

Let  a  very  sharp  needle  be  attached  to  any  smooth  insulated  metal 
body  provided  with  paper  or  pith-ball  indicators,  as  in  Fig.  225,  p.  222. 
Tf  the  body  is  now  charged  either  with  a  rubbed  rod  or  with  an  electric 
machine,  as  soon  as  'the  supply  of  electricity  is  stopped  the  paper  indi- 
cators will  immediately  fall,  showing  that  the  body  is  losing  its  charge. 
To  show  that  this  is  certainly  due  to  the  effect  of  the  point,  remove  the 
needle  and  repeat.  The  indicators  will  fall  very  slowly,  if  at  all. 

The  experiment  shows  that  the  electrical  density  upon  the 
point  is  so  great  that  the  charge  escapes  from  it  into  the  air. 
This  is  because  the  intense  charge  on  the  point  causes  many 
of  the  adjacent  molecules  of  the  air  to  lose  an  Electron.  This 
leaves  these  molecules  positively  charged.  The  free  electrons 
attach  themselves  to  neutral  molecules,  thus  charging  them 
negatively.  One  set  of  -these  electrically  charged  molecules 
(called  ions)  is  attracted  to  the  point  and  the  other  repelled 
away  from  it.  The  former  set  move  to  the  conductor,  give 
up  their  charges  to  it,  and  thus  neutralize  the  charge  upon  it. 

The  effect  of  points  may  be  shown  equally  well  by  charging  the  gold- 
leaf  electroscope  and  holding  a  needle  in  the  hand  within  a  few  inches 
of  the  knob.  The  leaves  will  fall  together  rapidly. 
In  this  case  the  needle  point  becomes  electrified 
by  induction  and  discharges  to  the  knob  electricity 
of  the  opposite  kind  to  that  on  the  knob,  thus 
neutralizing  its  charge.  An  entertaining  variation 
of  the  last  experiment  is  to  attach  a  tassel  of  tis- 
sue paper  to  an  insulated  conductor  and  electrify  it  Fi(,  ^  Discharg- 
strongly.  The  paper  streamers  under  their  mutual  mo.  effect  Of  points 
repulsions  will  stand  out  in  all  directions,  but  as 

soon  as  a  needle  point  is  held  in  the  hand  near  them,  they  will  at  once 
fall  together  (Fig.  232),  since  they  are  discharged  as  described  above. 

297.  The  electric  whirl.  Let  an  electric  whirl  (Fig.  233)  be  bal- 
anced upon  a  pin  point  and  attached  to  one  knob  of  an  electric  machine. 
As  soon  as  the  machine  is  started,  the  whirl  will  rotate  rapidly  in  the 
direction  of  the  arrows, 


230  STATIC  ELECTEICITY 

The  explanation  is  as  follows :  The  air  close  to  each  point 
is  ionized,  as  explained  in  §  296.  The  ions  of  sign  unlike 
that  of  the  charge  on  the  point  are  drawn  to  the  point  and 
discharged.  The  other  set 
of  ions  is  repelled.  But 
since  this  repulsion  is  mu- 
tual, the  point  is  pushed 
back  with  the  same  force 
with  which  these  ions  are 
pushed  forward ;  hence  the  FIG.  233.  The  FIG.  234.  The  elee- 

m,  TI     -i  •  electric  whirl  trie  wind 

rotation.    I  he  repelled  ions 

in  their  turn  drag  the  air  with  them  in  their  forward  motions, 
and  thus  produce  the  "electric  wind,"  which  may  be  detected 
easily  by  the  hand  or  by  a  candle  flame  (Fig.  234). 

298.  Lightning  and  lightning  rods.  It  was  in  1752  that 
Franklin,  during  a  thunderstorm,  sent  up  his  historic  kite. 
This  kite  was  provided  with  a  pointed  wire  at  the  top. 
As  soon  as  the  hempen  kite  string  had  become  wet  he  suc- 
ceeded in  drawing  ordinary  electric  sparks  from  a  key  at- 
tached to  the  lower  end.  This  experiment  demonstrated  for 
the  first  time  that  thunderclouds  carry  ordinary  electrical 
charges  which  may  be  drawn  from  them  by  points,  just  as  the 
charge  was  drawn  from  the  tassel  in  the  experiment  of  §  296. 
It  also  showed  that  lightning  is  nothing  but  a  huge  elec- 
tric spark.  Franklin  applied  this  discovery  in  the  invention 
of  the  lightning  rod.  The  way  in  which  the  rod  discharges 
the  cloud  and  protects  the  building  is  as  follows :  As  the 
charged  cloud  approaches  the  building  it  induces  an  opposite 
charge  in  the  rod.  This  induced  charge  escapes  rapidly  and 
quietly  from  the  sharp  point  in  the  manner  explained  above 
and  thus  neutralizes  the  charge  of  the  cloud. 

To  illustrate,  let  a  metal  plate  C  (Fig.  235)  be  supported  above  a 
metal  ball  E,  and  let  C  and  E  be  attached  to  the  two  knobs  of  an  electri- 
cal machine.  When  the  machine  is  started  sparks  will  pass  from  C  to  E, 


DISTRIBUTION  OF  CHARGE 


231 


but  if  a  point  p  is  connected  to  E,  the  sparking  will  cease ;  that  is,  the 
point  will  protect  E  from  the  discharges,  even  though  the  distance  Cp 
be  considerably  greater  than  CE. 

The  lower  end  of  a  lightning  rod  should  be  buried  deep 
enough  so  that  it  will  always  be  surrounded  by  moist  earth, 
since  dry  earth  is  a  poor  con- 
ductor. It  will  be  seen,  there- 
fore, that  lightning  rods  protect 
buildings  not  because  they  con- 
duct the  lightning  to  earth,  but 
because  they  prevent  the  for- 
mation of  powerful  charges  in 
the  neighborhood  of  the  build- 
ings on  which  they  are  placed. 

299.  Electric  screens.  That  the  charge  on  the  outside  of  a 
conductor  always  distributes  itself  in  such  a  way  that  there 
is  no  electric  force  within  the  conductor  was  first  proved 
experimentally  by  Faraday.  He  covered 
a  large  box  with  tin  foil  and  went  inside 
with  the  most  delicate  electroscopes  obtain- 
able. He  found  that  the  outside  of  the 
box  could  be  charged  so  strongly  that  long 
sparks  were  flying-  from  it  without  an/ 
electrical  effects  being  observable  anywhere 
inside  the  box. 


FIG.  235.    Illustrating  the  action  of 
a  lightning  rod 


FIG.  236.    Electro- 
scope protected  by 
a  wire  cage 


To  repeat  the  experiment  in  modified  form, 
let  an  electroscope  be  placed  beneath  a  bird  cage 
or  wire  netting,  as  in  Fig.  236.  Let  charged  rods  or  other  powerfully 
charged  bodies  be  brought  near  the  electroscope  outside  the  cage.  The 
leaves  will  be  found  to  remain  undisturbed. 

Hence,  if  we  wish  to  protect  an  electrical  instrument  from 
outside  electrical  disturbances,  we  have  only  to  surround  it 
with  a  metal  covering.* 

*  A  laboratory  exercise  on  static  electrical  effects  should  follow  the  discussion 
of  this  section.  See,  for  example,  Experiment  27  of  the  authors'  manual. 


232  STATIC  ELECTEICITY 

POTENTIAL  AND  CAPACITY 

300.  Potential  difference.  There  is  a  very  instructive  analogy 
between  the  use  of  the  word  "  potential "  in  electricity  and 
"  pressure  "  in  hydrostatics.  For  example,  if  water  will  flow 
from  tank  A  to  tank  B  through  the  connecting  pipe  R 
(Fig.  237),  we  infer  that  the  hydrostatic  pressure  at  a  must 
be  greater  than  that  at  5,  -and  we  attribute 
the  flow  directly  to  this  difference  in  pres- 
sure. In  exactly  the  same  way,  if,  when 
two  bodies  A  and  B  (Fig.  238)  are-con- 


nected  by  a  conducting  wire  r,  a  charge  FIG.  237.  Illustrating 
of  4-  electricity  is  found  to  pass  from  A  to  hydrostatic  pressure 
B,  that  is,  if  electrons  are  found  to  pass  from  B  to  J,  we  say  that 
the  electrical  potential  is  higher  at  A  than  at  B,  and  we  assign 
this  difference^  of  potential  as  the  cause  of  the  flow.*  Thus, 
just  as  water  tends  to  flow  from  points  of  higher  hydrostatic 
pressure  to  points  of  lower  hydrostatic  pressure,  so  electricity 
tends  to  flow  from  points  of  higher  electrical  pressure  or 
potential  to  points  of  lower  electrical  pressure  or  potential. 
Again,  if  water  is  not  continuously  supplied  to  one  of 
the  tanks  A  or  B  of  Fig.  237,  we  know  that  the  pressures 
at  a  and  b  must  soon  become  the 
same.  Similarly,  if  no  electricity  is 

supplied  to  the    bodies  A  and  B 

*   TV       noo     ^     -  *_•   i  FIG.  238.    Illustrating  electri- 

of  Fig.  238,  their  potentials  very  cal  pressure 

quickly  become  the  same.   In  other 

words,  all  points  on  a  system  of  connected  conductors  in  which 

the  electricity  is  in  a  stationary  or  static  condition  are  at  the  same 

*  Franklin  thought  that  it  was  the  positive  electricity  which  moved  through  a 
conductor,  while  he  conceived  the  negative  as  inseparably  associated  with  the 
atoms.  Hence  it  became  a  universally  recognized  convention  to  regard  electricity 
as  moving  through  a  conductor  in  the  direction  in  which  a  +  charge  would  have 
to  move  to  produce  the  observed  effect.  It  is  not  desirable  to  attempt  to  change 
this  convention  now  even  though  the  electron  theory  has  exactly  inverted  the 
roles  of  the  +  and  —  charges. 


POTENTIAL  AND  CAPACITY  233 

potential.  This  result  follows  at  once  from  the  fact  of  mobility 
of  electric  charges  through  conductors. 

But  if  water  is  continuously  poured  into  A  and  removed 
from  B  (Fig.  237),  the  pressure  at  a  will  remain  permanently 
above  the  pressure  at  b,  and  a  continuous  flow  of  water  will 
take  place  through  R.  So  if  A  (Fig.  238)  is  connected  with  an 
electrical  machine  and  B  to  earth,  a  permanent  potential  differ- 
ence will  exist  between  A  and  7?,  and  a  continuous  current  of 
electricity  will  flow  through  r.  Difference  in  potential  is 
commonly  denoted  simply  by  the  letters  P.D.  (Potential 
Difference). 

301.  Some  methods  of  measuring  potentials.  The  simplest 
and  most  direct  way  of  measuring  the  potential  difference  be- 
tween two  bodies  is  to  connect  one  to  the  knob,  the  other  to 
the  conducting  case,*  of  an  electroscope.  The  amount  of 
separation  of  the  gold  leaves  is  a  measure  of  the  P.D.  between 
the  bodies.  The  unit  in  which  P.D.  is  usually  expressed  is 
called  the  volt.  It  will  be  accurately  defined  in  §  331.  It  will 
be  sufficient  here  to  say  that  it  is  approximately  equal  to  the 
electrical  pressure  between  the  ends  of  a  strip  of  copper  and  a 
strip  of  zinc  immersed  in  dilute  sulphuric  acid  (see  Fig.  247). 

Since  the  earth  is  on  the  whole  a  good  conductor,  its  poten- 
tial is  everywhere  the  same  (§  300);  henc*e  it  makes  a  con- 
venient  standard  of  reference  in  potential  measurements.  To 
find  the  potential  of  a  body  relative  to  that  of  the  earth,  we 
connect  the  outer  case  of  the  electroscope  to  the  earth  by 
means  of  a  wire,  and  connect  the  body  to  the  knob.  If  the 
electroscope  is  calibrated  in  volts,  its  reading  gives  the  P.D. 
between  the  body  and  the  earth.  Such  calibrated  electroscopes 
are  called  electrostatic  voltmeters.  They  are  the  simplest  and  in 

*If  the  case  is  of  glass  it  should  always  be  made  conducting  by  pasting  tin-foil 
strips  on  the  inside  of  the  jar  opposite  the  leaves  and  extending  these  strips  over 
the  edge  of  the  jar  and  down  on  the  outside  to  the  conducting  support  on  which 
the  electroscope  regts.  The  object  of  this  is  to  maintain  the  walls  always  at  the 
potential  of  the  earth. 


234 


STATIC  ELECTRICITY 


many  respects  the  most  satisfactory  forms  of  voltmeters  to  be 

had.    Their  use,  both  in  laboratories  and  in  electrical  power 

plants,  is  rapidly  increasing.    They 

can  be  made  to  measure  a  P.D.  as 

small  as  1  0100  volt  and  as  large  as 

200,000  volts.    Fig.  239  shows  one 

of  the  simpler  forms.  The  outer  case 

is  of  metal  and  is  connected  to  earth 

at  the  point  a.    The    body  whose 

potential    is    sought    is    connected 

to  the  knob  b.    This  is  in  metallic 

contact  with   the   light  aluminium 

vane  <?,  which  takes  the  place  of  the 

gold  leaf. 

A  very  convenient  way  of  meas- 
uring a  large  P.D.  without  a  volt- 
meter is  to  measure  the  length  of  the 
spark  which  will  pass  between  the 
two  bodies  whose  P.D.  is  sought. 
The  P.D.  is  roughly  proportional  to  spark  length,  each  centi- 
meter of  spark  length  representing  a  P.D.  of  about  30,000  volts, 
if  the  electrodes  are  large  compared  to  their  distance  apart. 

302.  Condensers.  Let  a  metal  plate  A  be  mounted  on  an  insulating 
base  and  connected  with  an  electroscope,  as  in  Fig.  240.  Let  a  second 
plate  B  be  similarly  mounted  and  connected  to  the  earth  by  a  conducting 
wire.  Let  A  be  A  -B 

charged  and  the 
deflection  of  the 
gold  leaves  noted. 
If,  now,  we  push  B 
toward  A ,  we  shall 
observe  that  as  it 
comes  near,  the 


FIG.  239.    Electrostatic 
voltmeter 


FIG.  240.    The  principle  of  the  condenser 


leaves  begin  to  fall  together,  showing  that  the  potential  of  A  is  dimin- 
ished by  the  presence  of  B,  although  the  quantity  of  electricity  on  A  has 
remained  unchanged.  If  we  convey  additional  —  charges  to  A  with  the 


COUNT  ALESSANDRO  VOLTA  (1745-1827) 

Great  Italian  physicist,  professor  at  Como  and  at  Pavia ;  inventor 
of  the  electroscope,  the  electrophorus,  the  condenser,  and  the 
voltaic  pile  (a  form  of  galvanic  cell) ;  first  measured  the  potential 
differences  arising  from  the  contact  of  dissimilar  substances; 
ennobled  by  Napoleon  for  his  scientific  services;  the  volt,  the 
practical  unit  of  potential  difference,  is  named  in  his  honor 


POTENTIAL  AND  CAPACITY  235 

aid  of  a  proof  plane,  we  shall  find  that  many  times  the  original  amount 
of  electricity  may  now  be  put  on  A  before  the  leaves  return  to  their 
original  divergence  ;  that  is,  before  the  body  regains  its  original  potential. 

We  say,  therefore,  that  the  capacity  of  A  for  holding  elec- 
tricity has  been  very  greatly  increased  by  bringing  near  it 
another  conductor  which  is  connected  to  earth.  It  is  evident 
from  this  statement  that  we  measure  the  capacity  of  a  body  by 
the  amount  of  electricity  which  must  be  put  upon  it  to  raise  it  to 
a  given  potential.  The  explanation  of  the  increase  in  capacity 
in  this  case  is  obvious.  As  soon  as  B  was  brought  near  to  A 
it  became  charged,  by  induction,  with  electricity  of  opposite 
sign  to  A,  the  electricity  of  like  sign  to  A  being  driven  off  to 
earth  through  the  connecting  wire.  The  attraction  between 
these  opposite  charges  on  A  and  B  drew  the  electricity  on  A 
to  the  face  nearest  to  B  and  removed  it  from  the  more  remote 
parts  of  A,  so  that  it  became  possible  to  put  a  very  much 
larger  charge  on  A  before  the  tendency  of  the  electricity  on  A 
to  pass  over  to  the  electroscope  became  as  great  as  it  was  at 
first ;  that  is,  before  the  potential  of  A  rose  to  its  initial  value. 
In  such  a  condition  the  electricity  on  A  is  said  to  be  "  bound  " 
by  the  opposite  electricity  on  B. 

An  arrangement  of  this  sort  consisting  of  two  conductors  sepa- 
rated by  a  nonconductor  is  catted  a  condenser.  Tf  the  conducting 
plates  are  very  close  together  and  one  of  them  grounded,  the 
capacity  of  the  system  may  be  thousands  of  times  as  great  as 
that  of  one  of  the  plates  alone. 

303.  The  Leyden  jar.  The  most  common  form  of  condenser 
is  a  glass  jar  coated  part  way  to  the  top  inside  and  outside 
with  tin  foil  (Fig.  241).  The  inside  coating  is  connected  by  a 
chain  to  the  knob,  while  the  outside  coating  is  connected  to 
earth.  Condensers  of  this  sort  first  came  into  use  in  Leyden, 
Holland,  in  1745.  Hence  they  are  now  called  Leyden  jars. 

To  charge  a  Leyden  jar  the  outer  coating  is  held  in  the  hand  while 
^the  knob  is  brought  into  contact  with  one  terminal  of  an  electrical 


236 


STATIC  ELECTRICITY 


machine,  for  example  the  negative.  As  fast  as  electrons  pass  to  the 
knob  they  spread  to  the  inner  coat  of  the  jar,  where  they  repel  electrons 
from  the  outer  coat  to  the  earth,  thus  leaving  it  positively  charged.  If 
the  inner  and  outer  coatings  are  now  connected  by  a  discharging  rod, 
as  in  Fig.  241,  a  powerful  spark  will  be  produced.  Let  a  charged  jar 
be  placed  on  a  glass  plate  so  as  to  insulate 
the  outer  coat.  Let  the  knob  be  touched  with 
the  finger.  No  appreciable  discharge  will  be 
noticed.  Let  the  outer  coat  be  in  turn  touched 
with  the  finger.  Again  no  appreciable  discharge 
will  appear.  But  if  the  inner  and  outer  coatings 
are  connected  with  the  discharger,  a  powerful 
spark  will  pass. 


FIG.  241.   The  Leyden 
jar 


The  experiment  shows  that  it  is  im- 
possible to  discharge  one  side  of  the  jar 
alone,  for  practically  all  of  the  charge  is  bound  by  the  opposite 
charge  on  the  other  coat.  The  full  discharge  can  therefore 
occur  only  when  the  inner  and  outer  coats  are  connected. 


ELECTRICAL  GENERATORS 

304.  The  electrophorus.  The  electrophorus  is  a  simple  elec- 
trical generator  which  illustrates  well  the  principle  underlying 
the  action  of  all  electrostatic  machines.  All  such  machines 
generate  electricity  primarily  by  induction, 
not  by  friction.  B  (Fig.  242)  is  a  hard  rub- 
ber plate  which  is  first  charged  by  rubbing 
it  with  fur  or  flannel.  A  is  a  metal  plate 
provided  with  an  insulating  handle.  When 
the  plate  A  is  placed  upon  B,  touched  with 
the  finger,  and  then  removed,  it  is  found 
possible  to  draw  a  spark  from  it,  which  in 
dry  weather  may  be  a  quarter  of  an  inch 
or  more  in  length.  The  process  may  be  repeated  an  indefinite 
number  of  times  without  producing  any  diminution  in  the  size 
of  the  spark  which  may  be  drawn  from  A. 


FIG.  242.    The  elec- 
trophorus 


ELECTRICAL  GENERATORS  237 

If  the  sign  of  the  charge  on  A  is  tested  by  means  of  an  elec- 
troscope, it  will  be  found  to  be  positive.  This  proves  that  A 
has  been  charged  by  induction,  not  by  contact  with  B,  for  it 
is  to  be  remembered  that  the  latter  is  charged  negatively.  The 
reason  for  this  is  that  even  when  A  rests  upon  B  it  is  rh  reality 
separated  from  it,  at  all  but  a  very  few  points,  by  an  insulating 
layer  of  air ;  and,  since  B  is  a  nonconductor,  its  charge  cannot 
pass  off  appreciably  through  these  few  points  of  contact.  It 
simply  repels  negative  electricity  to  the  top  side  of  the  metal 
plate  A,  and  thus  charges  positively  the  lower  side.  The  nega- 
tive passes  off  to  earth  when  the  plate  is  touched  with  the 
finger.  Hence,  when  the  finger  is  removed  and  A  lifted,  it 
possesses  a  strong  positive  charge. 

305.  The  Toepler-Holtz  electrical  machine.  The  ordinary  static  ma- 
chine is  nothing  but  a  continuously  acting  electrophorus.  Fig.  243,  (1), 
represents  the  so-called  Toepler-Holtz  type  of  such  a  machine.  Upon 

(1) 

(2). 
R  S 


FIG.  243.    Toepler-Holtz  induction  machine 

the  back  of  the  stationary  plate  E  are  pasted  paper  sectors,  beneath 
which  are  strips  of  tin  foil  A  B  and  CD,  called  inductors.  In  front  of  E 
is  a  revolving  glass  plate  carrying  disks  I,  m,  n,  o,  p,  and  q,  called  carriers. 
To  the  inductors  AB  and  CD  are  fastened  metal  arms  t  and  u,  which 
bring  C  and  D  into  electrical  contact  with  the  disks  /,  m,  n,  o,  p,  and  q, 
when  these  disks  pass  beneath  the  tinsel  brushes  carried  by  t  and  u.  A 
stationary  metallic  rod  rs  carries  at  its  ends  stationary  brushes  as  well 
as  sharp-pointed,  metallic  combs.  The  two  knobs  R  and  S  have  their 
capacity  increased  by  the  Leyden  jars  L  and  U,  the  outer  coatings  of 
which  are  connected  beneath  the  base  of  the  machine. 


238 


STATIC  ELECTRICITY 


306.  Action  of  the  Toepler-Holtz  machine.    The  action  of  the  machine 
described  above  is  best  understood  from  the  diagram  of  Fig.  243,  (2). 
Suppose  that  a  small  +  charge  is  originally  placed  on  the  inductor  CD. 
Induction  takes  place  in  the  metallic  system  consisting  of  the  disks 
I  and  o  and  the  rod  rs,  I  becoming  negatively  charged  and  o  positively 
charged.  *  As  the  plate  carrying  /,  ?n,  n,  o,  p,  g  rotates  in  the  direction 
of  the  arrow  the  negative  charge  on  I  is  carried  over  to  the  position  m, 
where  a  part  of  it  passes  over  to  the  inductor  AB,  thus  charging  it 
negatively.  When  I  reaches  the  position  n,  the  remainder  of  its  charge, 
being  repelled  by  the  negative  which  is  now  on  AB,  passes  over  into 
the  Leyden  jar  L.   When  /  reaches  the  position  o,  it  again  becomes 
charged  by  induction,  this  time  positively,  and  more  strongly  than  at 
first,  since  now  the  negative  on  AB,  as  well  as  the  positive  on  CD,  is 
acting  inductively  upon  the  rod  rs.    When  I  reaches  the  position  u,  its 
now  strong  positive  charge  pulls  negative  from  CD,  thus  increasing  the 
positive  charge  upon  this  inductor.    In  the  position  v  negative  is  pulled 
out  of  U,  thus  leaving  it  positively  charged  and  discharging  I.    This 
completes  the  cycle  for  1.  Thus,  as  the  rotation  continues,  AB  and  CD 
acquire   stronger  and   stronger  charges,  the  inductive  action  upon  rs 
iSecomes  more  and  more  intense,  and  positive  and  negative  charges  are 
continuously  imparted  to  L'  and  L 

until  a  discharge  takes  place  between 
the  knobs  R  and  S. 

There  is  usually  sufficient  charge 
on  one  of  the  inductors  to  start  the 
machine,  but  in  damp  weather  it  will 
often  be  found  necessary  to  apply  a 
charge  to  one  of  the  inductors  before 
the  machine  will  start. 

307.  The  Wimshurst  electrical  ma- 
chine.    The  essential  difference  be- 
tween the  Toepler-Holtz  (Fig.  243) 
and   the  Wimshurst   electrical   ma- 
chine (Fig.  244)  is  that  the  latter  has 
two  plates  revolving  in  opposite  direc- 
tions, and  that  these  plates  carry  a 
large  number  of  tin-foil  strips  which 

act  alternately  as  inductors  and  as  carriers,  thus  dispensing  with  the 
necessity  of  separate  inductors.  The  action  of  the  machine  may  be 
understood  readily  from  Fig.  245.  Suppose  that  a  small  negative 
charge  is  placed  on  a.  This,  acting  inductively  on  the  rod  j*s,  charges 


FIG.  244.    The  Wimshurst  induc- 
tion machine 


ELECTEICAL  GEKEKATOBS  239 

a'  positively.  When  a'  in  the  course  of  the  rotation  reaches  the  posi- 
tion b',  it  acts  inductively  upon  the  rod  sY  and  thus  charges  the,  disk  b 
negatively.  It  will  be  seen  that  henceforth  all  the  disks  in  the  inner 
circle  receive  +  charges  as  they  pass  the  brush  r,  and  that  all  the 
disks  in  the  outer  circle,  that  is,  on  the 
back  plate,  receive  —  charges  as  they 
pass  the  brush  s'.  Similarly,  on  the 
lower  half  of  the  plates  all  the  disks  on 
the  inner  circle  receive  —  charges  as 
they  pass  the  brush  s,  and  all  the  disks 
on  the  outer  circle  receive  +  charges  as 
they  pass  the  brush  r'. 

When  the  positive  charges  on  the 

inner  disks  come  opposite   the    combs 

.  ,       FIG.  245.    Principle  of  Wniis- 

c,  they  are  transferred  to  the  +  knob  lmrgt  machine 

of  the  machine  or  to  the  Ley  den  jar 

connected  with  it.  The  same  process  is  occurring  on  the  other  side, 
where  —  charges  are  being  taken  off.  When  a  spark  passes,  the  Leyden 
jars  and  the  connecting  system  of  conductors  are  restored  to  their 
initial  conditions  and  the  process  begins  again. 


QUESTIONS  AND  PROBLEMS 

1.  With  a  stick  of  sealing  wax  and  a  piece  of  flannel,  in  what  two 
ways  could  you  give  a  positive  charge  to  an  insulated  body  ? 

2.  Will  a  solid  sphere  hold  a  larger  charge  of  electricity  than  a 
hollow  one  of  the  same  diameter? 

3.  When  a  negatively  electrified  cloud  passes  over  a  house  provided 
with  a  lightning  rod,  the  rod  discharges  positive  electricity  into  the 
cloud.    Explain. 

4.  Why  is  the  capacity  of  a  conductor  greater  when  another  con- 
ductor connected  to  the  earth  is  near  it  than  when  it  stands  alone? 

5.  A  Leyden  jar  is  placed  on  a  glass  plate  and  10  units  of  electricity 
placed  on  the  inner  coating.    The  knob  is  then  connected  to  a  gold-leaf 
electroscope.    Will  the  leaves  of  the  electroscope  stand  farther  apart  now 
or  after  the  outside  coating  has  been  connected  to  the  earth  ? 

6.  Why  cannot  a  Leyden  jar  be  appreciably  charged  if  the  outer 
coat  is  insulated? 

7.  Why  is  it  not  necessary  to  connect  to  earth  the  outer  coatings 
of  the  Leyden  jars  on  an  electrical  machine  to  charge  them  fully,  pro- 
vided they  are  connected  to  one  another? 


CHAPTER  XIII 


ELECTRICITY  IN  MOTION* 

DETECTION  AND  MEASUREMENT  OF  ELECTRIC  CURRENTS 
308.  Electricity  in  motion  produces  a  magnetic  effect.    Let  a 

powerfully  charged  Leyderi  jar  be  discharged  through  a  coil  which  sur- 
rounds an  unmagnetized  knitting  needle,  insulated  by  a  glass  tube,  in 
the  manner  shown  in  Fig.  246. 
After  the  discharge  the  needle 
will  be  found  to  be  distinctly 
magnetized.  If  the  sign  of  the 
charge  on  the  jar  is  reversed,  the 
poles  will  in  general  be  reversed. 

The  experiment  shows  that 
there  is  a  definite  connection 
between  electricity  and  mag- 
netism. Just  what  this  con- 
nection is  we  do  not  yet  know 
with  certainty,  but  we  do  know  that  magnetic  effects  are  always 
observable  near  the  path  of  a  moving  electrical  charge,  while 
no  such  effects  can  ever  be  observed  near  a  charge  at  rest. 

To  prove  that  a  charge  at  rest  does  not  produce  a  magnetic  effect, 
let  a  charged  body  be  brought  near  a  compass  needle.  It  will  attract 
either  end  of  the  needle  with  equal  readiness.  While  the  needle  is 
deflected,  insert  between  it  and  the  charge  a  sheet  of  zinc,  aluminium, 
brass,  or  copper.  This  will  act  as  an  electric  screen  (see  §  299,  p.  231) 
and  will  therefore  cut  off  all  effect  of  the  charge.  The  compass  needle 
will  at  once  swing  back  to  its  north-and-south  position. 

*  This  chapter  should  be  accompanied  or,  better,  preceded!  by  laboratory  experi- 
ments on  the  simple  cell  and  on  the  magnetic  effects  of  a  current.  See,  for  exam- 
ple, Experiments  28,  29,  and  30  of  the  authors'  manual. 

240 


FIG.  246.    Magnetizing  effect  of  spark 
on  knitting  needle 


MEASUREMENT  OF  ELECTRIC  CURRENTS      241 

Let  the  compass  needle  be  deflected  by  a  bar  magnet  and  let  the 
screen  be  inserted  again.  The  sheet  of  metal  does  not  cut  off  the  mag- 
netic forces  in  the  slightest  degree. 

The  fact  that  an  electric  charge  exerts  no  magnetic  force  is  shown, 
then,  both  by  the  fact  that  it  attracts  either  end  of  the  compass  needle 
with  equal  readiness,  and  also  by  the  fact  that  the  screen  cuts  off  its 
action  completely,  while  the  same  screen  does  not  have  any  effect  in 
cutting  off  the  magnetic  force. 

An  electrical  charge  in  motion  is  called  an  electric  current, 
and  its  presence  is  most  commonly  detected  by  the  magnetic 
effect  which  it  produces. 

309.  The  galvanic  cell.  When  a  Leyden  jar  is  discharged, 
but  a  very  small  quantity  of  electricity  passes  through  the  con- 
necting wires,  since  the  current  lasts  for  but  a  small  fraction 
of  a  second.  If  we  could  keep  a  current  flow- 
ing continuously  through  the  wire,  we  should 
expect  the  magnetic  effect  to  be  much  more 
pronounced.  It  was  in  1786  that  Galvani,  an 
Italian  anatomist  at  the  University  of  Bologna, 
accidentally  discovered  that  there  is  a  chemical 

method  for  producing  such  a  continuous  cur-     *IG-247-  Snn- 
5  ,  ,    ,  pie  voltaic  cell 

rent.  His  discovery  wads  not  understood,  how- 
ever, until  Volta,  while  endeavoring  to  throw  light  upon  it, 
in  1800  invented  an  arrangement  which  is  now  known 
sometimes  as  the  voltaic  and  sometimes  as  the  galvanic  cell. 
This  consists,  in  its  simplest  form,  of  a  strip  of  copper  and 
a  strip  of  zinc  immersed  in  dilute  sulphuric  acid  (Fig.  247). 

Let  the  terminals  of  such  a  cell  be 
connected  for  a  few  seconds  to  the 
ends  of  the  coil  of  Fig.  246  when  an 
unmagnetized  needle  lies  within  the 
glass  tube.  The  needle  will  be  found 
to  have  become  magnetized  much  FIQ>  24g  Oergted,8  experiment 
more  strongly  than  before.  Again, 

let  the  wire  which  connects  the  terminals  of  the  cell  be  held  above  a 
magnetic  needle,  as  in  Fig.  248  ;  the  needle  will  be  strongly  deflected. 


242  ELECTRICITY  IN  MOTION 

Evidently,  then,  the  wire  which  connects  the  terminals  of  a 
galvanic  cell  carries  a  current  of  electricity.  Historically  the 
second  of  these  experiments,  performed  by  the  Danish  physicist 
Oersted  in  1820,  preceded  the  discovery  of  the  magnetizing 
effects  of  currents  upon  needles.  It  created  a  great  deal  of 
excitement  at  the  time,  because  it  was  the  first  clue  which 
had  been  found  to  a  relationship  between  electricity  and 
magnetism. 

310.  Plates  of  a  galvanic  cell  are  electrically  charged.  Since 
an  electric  current  flows  through  a  wire  as  soon  as  it  is  touched 
to  the  zinc  and  copper  strips  of  a  galvanic  cell,  we  at  once 
infer  that  the  terminals  of  such  a  cell  are  electrically  charged 
before  they  are  connected.  That  this  is  indeed  the  case  may 
be  shown  as  follows : 

Let  a  metal  plate  A  (Fig.  249),  covered  with  shellac  on  its  lower  side 
and  provided  with  an  insulating  handle,  be  placed  upon  a  similar  plate 
B  which  is  in  contact  with  the  knob  of  an  electroscope.  Let.  the  copper 
plate  of  a  galvanic  cell  be  connected  with  A  and  the  zinc  plate  with  B, 
as  in  Fig.  249.  Then  let  the  connecting  wires 
be  removed  and  the  plate  A  lifted  away  from 
B.  The  opposite  electrical  charges  which  were 
bound  by  their  mutual  attractions  to  the  adja- 
cent faces  of  A  and  B,  so  long  as  these  faces 
were  separated  only  by  the  thin  coat  of  shellac, 
are  freed  as  soon  as  A  is  lifted,  and  hence  part 
of  the  charge  on  B  passes  to  the  leaves  of  the 

electroscope.    These  leaves  will  indeed  be  seen        FlG*  249'     ShowinS 
•       T*          i       .,          -i      i  •  -i    !        i  charges  on  plates  of 

to  diverge.    If  an  ebonite  rod  which  has  been  It          n 

rubbed  with  flannel  or  cat's  fur  is  brought  near 

the  electroscope,  the  leaves  wTill  diverge  still  farther,  thus  showing  that 
the  zinc  plate  of  the  galvanic  cell  is  negatively  charged.*  If  the  experi- 
ment is  repeated  with  the  copper  plate  in  contact  with  B  and  the  zinc 
in  contact  with  A,  the  leaves  will  be  found  to  be  positively  charged. 

*  If  the  deflection  of  the  gold  leaves  is  too  small  for  purposes  of  demonstration, 
let  a  hattery  of  from  five  to  ten  cells  be  used  instead  of  the  single  cell.  However, 
if  the  plates  A  and  B  are  three  or  four  inches  in  diameter,  and  if  their  surfaces 
are  very  flat,  a  single  cell  is  sufficient. 


HANS  CHRISTIAN  OERSTED 

(1777-1851) 

The  discoverer  of  the  connection 
betAveen  electricity  and  magnetism 
was  a  Dane  and  a  professor  at  the 
University  of  Copenhagen.  His 
famous  experiment  made  in  1820 
stimulated  the  researches  which 
led  to  the  modern  industrial  devel- 
opments of  electricity 


JOSEPH  HENRY  (1797-1878) 

Born  in  Albany,  New  York  ;  taught 
physics  and  mathematics  in  Albany 
Academy  anrf  Princeton  College. 
He  invented  the  electromagnet 
(1828),  discovered  the  oscillatory 
nature  of  the  electric  spark  (1842) 
by  magnetizing  needles  in  the 
manner  described  on  page  241,  and 
made  the  first  experiments  in  self- 
induction  (1&32).  He  was  the  first 
secretary  of  the  Smithsonian  Insti- 
tution, and  the  organizer  of  the 
Weather  Bureau 


MEASUREMENT  OF  ELECTRIC  CURRENTS      243 

The  terminals  of  a  galvanic  cell  therefore  carry  positive 
and  negative  charges  just  as  do  the  terminals  of  an  electrical 
machine  in  operation.  The  +  charge  is  always  found  upon  the 
copper  and  the  —  charge  upon  the  zinc.  The  source  of  these 
charges  is  the  chemical  action  which  takes  place  within  the 
cell.  When  these  terminals  are  connected  by  a  conductor  a 
current  flows  through  the  latter  just  as  in  the  case  of  the 
electrical  machine,  and  it  is  the  universal  custom  to  consider 
that  it  flows  from  positive  to  negative  (see  §  300  and  footnote), 
that  is,,  from  copper  to  zinc. 

311.  Comparison  of  a  galvanic  cell  and  static  machine.  If 
one  of  the  terminals  of  a  galvanic  cell  is  touched  directly  to 
the  knob  of  a  gold-leaf  electroscope,  without  the  use  of  the 
condenser  plates  A  and  B  of  Fig.  249,  no  divergence  of  the 
leaves  will  be  detected ;  but  if  one  knob  of  the  static  machine 
in  operation  were  so  touched,  the  leaves  would  probably  be 
torn  apart  by  the  violence  of  the  divergence.  Since  we  have 
seen  in  §  301  that  the  divergence  of  the  gold  leaves  is  a  meas- 
ure of  the  potential  of  the  body  to  which  they  are  connected, 
we  learn  from  this  experiment  that  the  chemical  actions  in  the 
galvanic  cell  are  able  to  produce  between  its  terminals  but  a 
very  small  potential  difference  in  comparison  with  that  pro- 
duced by  the  static  machine  between  its  terminals.  As  a  matter 
of  fact  the  potential  difference  between  the  terminals  of  the 
cell  is  about  one  volt,  while  that  between  the  knobs  o£  the 
electrical  machine  may  be  as  much  as  200,000  volts. 

But  if  the  knobs  of  the  static  machine  are  connected  to  the 
ends  of  the  wire  of  Fig.  248,  and  the  machine  operated,  the  cur- 
rent sent  through  the  wire  will  not  be  large  enough  to  produce 
any  appreciable  effect  upon  the  needle.  Since  under  these  same 
circumstances  the  galvanic  cell  produced  a  very  large  effect 
upon  the  needle,  we  learn  that  although  the  cell  develops  a  very 
small  P.D.  between  its  terminals,  it  nevertheless  sends  through 
the  connecting  wire  very  much  more  electricity  per  second 


244 


ELECTRICITY  IN  MOTION 


than  the  static  machine  is  able  to  send.  This  is  because  the' 
chemical  action  of  the  cell  is  able  to  recharge  the  plates  to 
their  small  P.D.  practically  as  fast  as  they  are  discharged 
through  the  wire,  whereas  the  static  machine  requires  a  rela- 
tively long  time  to  recharge  its  terminals  to  their  high  P.D. 
after  they  have  been  once  discharged. 

312.  Shape  of  the  magnetic  field  about  a  current.  If  we  place 
the  wire  which  connects  the  plates  of  a  galvanic  cell  in  a  vertical  posi- 
tion [Fig.  250,  (1)]  and  explore  with  a  compass  needle  the  shape  of  the 

(2) 


FIG.  250.    Magnetic  field  about  a  current 


magnetic  field  about  the  current,  we  find  that  the  magnetic  lines  are 
concentric  circles  lying  in  a  plane  perpendicular  to  the  wire  and  having 
the  wire  as  their  common  center.  If  we  reverse  the  direction  of  the 
current,  we  find  that  the  direction  in  which  the  compass  needle  points 
reverses  also.  If  the  current  is  very  strong  (say  40  amperes),  this  shape 
of  the  field  can  be  shown  by  scattering  iron  filings  on  a  plate  through 
which  the  current  passes,  in  the  manner  shown  in  Fig.  250,  (1).  If  the 
current  is  weak  the  experiment  should  be  performed  as  indicated  in 
Fig.  250,  (2). 

The  relation  between  the  direction  in  which  the  current 
flows  and  the  direction  in  which  the  N  pole  of  the  needle 
points  (this  is,  by  definition,  the  direction  of  the  magnetic 
field)  is  given  in  the  following  convenient  rule :  If  the  right 


MEASUREMENT  OF  ELECTRIC  CURRENTS      245 


FIG.  251.   The  right-hand  rule 


hand  grasps  the  wire  as  in  Fig.  251,  so  that  the  thumb  points  in 
the  direction  in  which  the  current  is  flowing,  then  the  magnetic 
lines  encircle  the  wire  in  the 
same  direction  as  do  the  fingers 
of  the  hand. 

313.  The  measurement  of  elec- 
trical currents.  Electrical  cur- 
rents are,  in  general,  measured  by  the  strength  of  the  magnetic 
effect  which  they  are  able  to  produce  under  specific  conditions. 
Thus,  if  the  wire  carry- 
ing a  current  is  wound 
into  circular  form  as 
in  Fig.  252,  the  right- 
hand  rule  shows  us 
that  the  shape  of  the 
magnetic  field  at  the 

center    of   the  coil   is 

.     ..  ,,          ,  FIG.  252.   Magnetic  field  about  a  circular  coil 

similar  to  that  shown  carrying  a  current 

in  the  figure.   If,  then, 

the  coil  is  placed  in  a  north-and-south  plane  and  a  compass 
needle  is  placed  at  the  center,  the  passage  of  the  current 
through  the  coil  tends  to  deflect  the  needle 
so  as  to  make  it  point  east  and  west.  TKe 
amount  of  deflection  under  these  conditions 
is  taken  as  the  measure  of  current  strength. 
The  unit  of  current  is  called  the  ampere  and 
is  in  fact  approximately  the  same  as  the  cur- 
rent which,  flowing  through  a  circular  coil 
of  three  turns  and  10  centimeters  radius,  set 
in  a  north-and-south  plane,  will  produce  a 
deflection  of  45  degrees  at  Washington  in  a 
small  compass  needle  placed  at  its  center. 
The  legal  definition  of  the  ampere  is,  however,  based  on 
the  chemical  effect  of  a  current.  It  will  be  given  in  §  339. 


FIG.  253.     Simple 
suspended-coil  gal- 
vanometer 


246 


ELECTEICITY  IN  MOTION 


Nearly  all  current-measuring  instruments  consist  essentially 
either  of  a  small  compass  needle  at  the  center  of  a  fixed  coil 
as  in  Fig.  252,  or  of  a  movable  coil 
suspended  between  the  poles  of  a 
fixed  magnet  in  the  manner  illus- 
trated roughly  in  Fig.  253.  The  pas- 
sage of  the  current  through  the  coil 
produces  a  deflection,  in  the  first  case, 
of  the  magnetic  needle  with  refer- 
ence to  the  fixed  coil,  and  in  the 
second  case,  of  the  coil  with  reference 
to  the  fixed  magnet.  If  the  instru- 
ment has  been  calibrated  to  give 
the  strength  of  the  current  directly 

in  amperes,  it  is  called  an  ammeter,      Fm   254    A  lecturc.table 
otherwise  a  galvanometer  (Fig.  254).  galvanometer 

QUESTIONS  AND  PROBLEMS 

1.  How  could  you  test  whether  or  not  the  strength  of  an  electric 
current  is  the  same  in  all  parts  of  a  circuit  ?    Try  it. 

2.  Under  what  conditions  will  an  electric  charge  produce  a  magnetic 
effect? 

3.  In  what  direction  will  the  north  pole  of  a  magnetic  needle  be 
deflected  if  it  is  held  above  a  current  flowing  from  north  to  south  ? 

4.  A  man  stands  beneath  a  north-and-south  trolley  line  and  finds  that 
a  magnetic  needle  in  his  hand  has  its  north  pole  deflected  toward  the 
east.    What  is  the  direction  of  the  current  flowing  in  the  wire  ? 

5.  A  loop  of  wire  lying  on  the  table  carries  a  current  which  flows 
around  it  in  clockwise  direction.    Would  a  north  magnetic  pole  at  the 
center  of  the  loop  tend  to  move  up  or  down  ? 

6.  When  a  compass  needle  is  placed,  as  in  Fig.  252,  at  the  middle  of 
a  coil  of  wire  which  lies  in  a  north-and-south  plane,  the  deflection  pro- 
duced in  the  needle  by  a  current  sent  through  the  coil  is  approximately 
proportional  to  the  strength  of  the  current,  provided  the  deflection  is 
small  —  not  more,  for  example,  than  20°  or  25° ;  but  when  the  deflection 
becomes  large  —  say  60°  or  70°  —  it  increases  very  much  more  slowly 
than  does  the  current  which  produces  it.    Can  you  see  any  reason  why 
this  should  be  so? 


J 


ANDRE  MARIE  AMPERE  (1775-1836) 

French  physicist  and  mathematician;  son  of  one  of  the  victims 
of  the  guillotine  in  1793 ;  professor  at  the  Polytechnic  School  in 
Paris  and  later  at  the  College  of  France ;  began  his  experiments 
on  electromagnetism  in  1820,  very  soon  after  Oersted's  discovery  ; 
published  his  great  memoir  on  the  magnetic  effects  of  currents 
in  1823 ;  first  stated  the  rule  for  the  relation  between  the  direction 
of  a  current  in  a  wire  and  the  direction  of  the  magnetic  field 
about  it.  The  ampere,  the  practical  unit  of  current,  is  named 
in  his  honor 


ELECTROMOTIVE  FORCE  AND  RESISTANCE    247 


ELECTROMOTIVE  FORCE  AND  RESISTANCE 

314.  Electromotive  force  and  its  measurement.*  The  poten- 
tial difference  which  a  galvanic  cell  or  any  other  generator  of 
electricity  is  able  to  maintain  between  its  terminals  when 
these  terminals  are  not  connected  by  a  wire,  that  is,  the  total 
electrical  pressure  which  the  generator  is  capable  of  exerting, 
is  commonly  called  its  electromotive  force,  usually  abbreviated 
to  E.M.F.  The  E.M.F.  of  an  electrical  generator  may  then  be 
defined  as  its  capacity  for  producing  electrical  pressure,  or  P.D. 
This  P.D.  might  be  measured,  as  in  §  301, 
by  the  deflection  produced  in  an  electro- 
scope when  one  terminal  was  connected  to 
the  case  of  the  electroscope  and  the  other 
terminal  to  the  knob.  Potential  differ- 
ences are  in  fact  measured  in  this  way  in 
all  so-called  electrostatic  voltmeters. 

The  more  common  type  of  potential- 
difference  measurer  consists,  however,  of 
an  instrument  made  like  a  galvanometer 
(Fig.  254),  save  that  the  coil  of  wire  is 
made  of  very  many  turns  of  extremely 
fine  wire,  so  that  it  carries  a  very  small 
current.  The  amount  of  current  which  it  does  carry,  however, 
is  proportional  to  the  difference  in  electrical  pressure  existing 
between  its  ends  when  these  are  touched  to  the  two  points 
whose  P.D.  is  sought.  The  principle  underlying  this  type  of 
voltmeter  will  be  better  understood  from  a  consideration  of 
the  following  water  analogy.  If  the  stopcock  K  (Fig.  255) 
in  the  pipe  connecting  the  water  tanks  C  and  D  is  closed,  and 
if  the  water  wheel  A  is  set  in  motion  by  applying  a  weight  W, 
the  wheel  will  turn  until  it  creates  such  a  difference  in  the 

*  This  subject  should  be  preceded  or  accompanied  by  laboratory  work  on 
E.M.F.   See,  for  example,  Experiment  31  of  the  authors'  manual, 
t 


FIG.  255.    Hydrostatic 

analogy  of  the  action 

of  a  galvanic  cell 


248  ELECTEICITY  IN  MOTION 

water  levels  between  C  and  D  that  the  back  pressure  against 
the  left  face  of  the  wheel  stops  it  and  brings  the  weight  W  to 
rest.  In  precisely  the  same  way  the  chemical  action  within 
the  galvanic  cell  whose  terminals  are  not  joined  (Fig.  256) 
develops  positive  and  negative  charges  upon  these  terminals; 
that  is,  creates  a  P.D.  between  them,  until  the  back  electrical 
pressure  through  the  cell  due  to  this  P.D.  is  sufficient  to  put 
a  stop  to  further  chemical  action.  The  seat  of  the  E.M.F.  is 
at  the  surfaces  of  contact  of  the  metals  with  the  acid,  where 
the  chemical  actions  take  place.  The  E.M.F.  of  the  cell  has  its 
water  analogy  in  'the  wheel  A,  which  creates 
the  difference  in  water  level  between  C  and  D. 

Now,  if  the  water  reservoirs  (Fig.  255)  are 
put  in   communication   by  opening   the   stop- 
cock If,  the  difference  in  level  between  C  and 
D  will  begin  to  fall,  and  the  wheel  will  begin 
to  build  it  up  again.   But  if  the  carrying  capac- 
ity of  the  pipe  ab  is  small  in  comparison  with     FIG  25(.   M ,,.,,._ 
the   capacity   of  the  wheel  to  remove  water     urementof.iM). 
from  i>  and  to  supply  it  to  <7,  then  the  differ-     between  the  ter- 
ence  of  level  which  permanently  exists  between      ^^anic  celf  ^ 
C  and  D  when  K  is  open  will  not  be  appreciably 
smaller  than  when  it  is  closed.    In  this  case  the  current  which 
flows  through  AB  may  obviously  be  taken  as  a  measure  of 
the  difference  in  pressure  which  the  pump  is  able  to  maintain 
between  C  and  D  when  K  is  closed. 

In  precisely  the  same  way,  if  the  terminals  C  and  D  of  the 
cell  (Fig.  256)  are  connected  by  attaching  to  them  the  ter- 
minals a  and  b  of  any  conductor,  they  at  once  begin  to  dis- 
charge through  this  conductor,  and  their  P.D.  therefore  begins 
to  fall.  But  if  the  chemical  action  in  the  cell  is  able  to  recharge 
C  and  D  very  rapidly  in  comparison  with  the  ability  of  the 
wire  to  discharge  them,  then  the  P.D.  between  C  and  D  will 
not  be  appreciably  lowered  by  the  presence  of  the  connecting 


ELECTROMOTIVE  FORCE  AND  RESISTANCE     249 


conductor.  In  tins  case  the  current  which  flows  through  the 
conducting  coil,  and  therefore  the  deflection  of  the  needle  at 
its  center,  may  be  taken  as  a  measure  of  the  electrical  pres- 
sure developed  by  the  cell;  that  is,  of  the  P.D.  between  its 
unconnected  terminals. 

The  common  voltmeter  is,  then,  exactly  like  an  ammeter, 
save  that  it  offers  so  high  a  resistance  to  the  passage  of  elec- 
tricity through  it  that  it  does  not  appreciably  reduce  the  P.D. 
between  the  points  to  which  it  is  connected. 

To  determine  experimentally  whether  or  not  touching  the  ends  of 
any  particular  galvanometer  to  the  terminals  of  a  cell  does  appreciably 
lower  their  P.D.,  let  the  galvanometer  in  question*  be  connected 
directly  to  the  terminals  of  the  cell  and  the  deflec- 
tion noted ;  then  let  the  ends  of  a  second  coil  of 
wire  which  has  exactly  the  same  carrying  capacity 
as  the  galvanometer  coil  be  also  touched  to  the 
terminals,  the  galvanometer  coil  being  still  in  cir- 
cuit. If  the  second  coil  has  sufficient  carrying 
capacity  to  appreciably  discharge  the  terminals,  the 
deflection  of  the  needle  of  the  galvanometer  will  be 
instantly  diminished  when  the  ends  of  the  second 
coil  are  brought  into  contact  with  them.  If  no  such 
diminution  is  observed,  we  may  know  that  the  second 
coil  does  not  discharge  the  terminals  of  the  cell  fas£ 

enough  to  appreciably  lower  their  P.D.,  and  hence  that  the  introduction 
of  the  first  coil,  which  was  of  equal  carrying  capacity,  also  did  not 
appreciably  lower  the  P.D.  between  the  terminals.  To  show  that  a  coil 
of  greater  carrying  capacity  will  at  once  lower  the  P.D.  between  C  and 
D  as  soon  as  it  is  touched  across  them,  let  a  coil  of  thicker  wire  be  so 
touched.  The  deflection  of  the  needle  will  be  diminished  instantly. 

315.  The  electromotive  forces  of  galvanic  cells.  Let  a  voltmeter 

of  any  sort  be  connected  to  the  terminals  of  a  simple  galvanic  cell,  like 
that  of  Fig.  247.  Then  let  the  distance  between  the  plates  and  the 

*  A  vertical  lecture-table  voltmeter  (Fig.  257)  and  a  similar  ammeter  are  desir- 
able for  this  and  some  of  the  following  experiments,  but  homemade  high-  and 
low-resistance  galvanometers,  like  those  described  in  the  authors'  manual,  are 
thoroughly  satisfactory,  save  for  the  fact  that  one  student  must  take  the  readings 
for  the  class. 


FIG.  257.    Lecture- 
table  voltmeter 


250 


ELECTRICITY  IN  MOTION 


amount  of  their  immersion  be  changed  through  wide  limits.    It  will  be 

found  that  the  deflection  produced  is  altogether  independent  of  the 

shape  or  size  of  the  plates  or  their  distance 

apart.    But  if  the  nature  of  the  plates  is 

changed,  the  deflection  changes.  Thus,  while 

copper    and    zinc    in    dilute    sulphuric    acid 

have  an    E.M.F.  of    one    volt,  carbon    and 

zinc  show  an  E.M.F.  of  at  least  1.5  volts, 

while    carbon    and    copper    wTill    show    an 

E.M.F.    of    very    much    less    than    a    volt. 

Similarly,   by   changing  the   nature   of   the 

liquid  in  which  the  plates   are  immersed, 

we  can  produce  changes  in  the  deflection  of 

the  voltmeter. 


FIG.  258.  Showing  method 
of  connecting  voltmeter 
to  find  P.D.  between  any 
two  points  m  and  n  on  an 
electrical  circuit 


We  learn  therefore  that  the  E.M.F. 
of  a  galvanic  cell  depends  simply  upon 
the  materials  of  which  the  cell  is  com- 
posed and  not  at  all  upon  the  shape, 
size,  or  distance  apart  of  the  plates. 

316.  Fall  of  potential  along  a  conductor  carrying  a  current. 
Not  only  does  a  P.D.  exist  between  the  terminals  of  a  cell  on 
open  circuit,  but  also  between  any  two  points  on  a  conductor 
through  which  a  current  is  passing. 
For  example,  in  the  electrical  circuit 
shown  in  Fig.  258  the  potential  at 
the  point  a  is  higher  than  that  at  m, 
that  at  m  higher  than  that  at  n,  etc., 
just  as  in  the  water  circuit  shown  in 
Fig.  259,  the  hydrostatic  pressure  at 
a  is  greater  than  that  at  m,  that  at  m 
greater  than  that  at  n,  etc.  The  fall 
in  the  water  pressure  between  m  and 
n  (Fig.  259)  is  measured  by  the  water 
head  n's.  If  we  wish  to  measure  the 
fall  in  electrical  potential  between 
m  and  n  (Fig.  258),  we  touch  the 


™.—. 

5  —  ._ 

rt 

—  — 

9'... 

FIG.  259. 


DD 

Hydrostatic  anal- 


ogy of  fall  of  potential  in  an 
electrical  circuit 


ELECTEOMOTIVE  FORCE  AND  RESISTANCE    251 

terminals  of  a  voltmeter  to  these  points  in  the  manner  shown 
in  the  figure.  Its  reading  gives  us  at  once  the  P.D.  between 
m  and  n  in  volts,  provided  always  that  its  own  current-carrying 
capacity  is  so  small  that  it  does  not  appreciably  lower  the  P.D. 
between  the  points  m  and  n  by  being  touched  across  them; 
that  is,  provided  the  current  which  flows  through  it  is  negli- 
gible in  comparison  with  that  which  flows  through  the  con- 
ductor which  already  joins  the  points  m  and  n. 

317.  Electrical  resistance.*  Let  the  circuit  of  a  galvanic  cell  be 
connected  through  a  lecture-table  ammeter,  or  any  low-resistance  gal- 
vanometer, and,  for  example,  20  feet  of  No.  30  copper  wire,  and  let  the 
deflection  of  the  needle  be  noted.  Then  let  the  copper  wire  be  replaced 
by  an  equal  length  of  No.  30  German-silver  wire.  The  deflection  will 
be  found  to  be  a  v$ry  small  fraction  of  what  it  was  at  first. 

A  cell,  therefore,  which  is  capable  of  developing  a  certain 
fixed  electrical  pressure  is  able  to  force  very  much  more  cur- 
rent through  a  given  wire  of  copper  than  through  an  ex- 
actly similar  wire  of  German  silver.  We  say,  therefore, 
that  German  silver  offers  a  higher  resistance  to  the  passage 
of  electricity  than  does  copper.  Similarly,  every  particular 
substance  has  its  own  characteristic  power  of  transmitting 
electrical  currents.  Since  silver  is  the  best  conductor  known, 
resistances  of  different  substances  are  commonly  referred  to 
it  as  a  standard,  and  the  ratio  between  the  resistance  of  a 
given  wire  of  any  substance  and  the  resistance  of  an  exactly 
similar  silver  wire  is  called  the  specific  resistance  of  that  sub- 
stance. The  specific  resistances  of  some  of  the  commoner 
metals  in  terms  of  silver  are  given  below : 

Silver     .    .    .    1.00  Soft  iron    .    .    6.00  German  silver  .    14-20 

Copper  .    .    .    1.11  Nickel    .    .    .    4.67  Hard  steel     .    .    13.5 

Aluminum     .    1.87  Platinum    .    .    7.20  Mercury    .    .    .    63.1 

*This  subject  should  be  accompanied  and  followed  by  laboratory  experiments 
on  Ohm's  law,  on  the  comparison  of  wire  resistances,  and  on  the  measurement  of 
internal  resistances.  See,  for  example,  Experiments  32,  33,  and  34  of  the  authors' 
manual. 


252  ELECTRICITY  IN  MOTION 

The  resistance  of  any  conductor  is  directly  proportional  to 
its  length  and  inversely  proportional  to  the  area  of  its  cross 
section. 

The  unit  of  resistance  is  the  ohm,  so  called  in  honor  of  the 
great  German  physicist,  Georg  Ohm  (1789-1854).  A  length 
of  9.35  feet  of  No.  30  copper  wire,  or  6.2  inches  of  No.  30 
German-silver  wire,  has  a  resistance  of  about  one  ohm.  The 
legal  definition  of  the  ohm  is  a  resistance  equal  to  that  of  a  column 
of  mercury  106.3  centimeters  long  and  1  square  millimeter  in  cross 
section,  at  0°  0. 

318.  Resistance  and  temperature.   Let  the  circuit  of  a  galvanic 
cell  be  closed  through  a  very  low-resistance  galvanometer  and  about 
10  feet  of  No.  30  iron  wire  wrapped  about  a  strip  of  asbestos.    Let  the 
deflection  of  the  galvanometer  be  observed  as  the  wire  is  heated  in  a 
Bunsen  flame.    As  the  temperature  rises  higher  and  higher  the  current 
will  be  found  to  fall  continually. 

The  experiment  shows  that  the  resistance  of  iron  increases 
with  rising  temperature.  This  is  a  general  law  which  holds  for 
all  metals.  In  the  case  of  liquid  conductors,  on  the  other  hand, 
the  resistance  usually  decreases  with  increasing  temperature. 
Carbon  and  a  few  other  solids  show  a  similar  behavior,  the 
filament  in  an  incandescent  electric  lamp  having  only  about 
half  the  resistance  when  hot  which  it  has  when  cold. 

319.  Ohm's  law.    In  1826  Ohm  announced  the  discovery 
that  the  currents  furnished  by  different  galvanic  cells,  or  combi- 
nations of  cells,  are  always  directly  proportional  to  the  E.M.F.^s 
existing  in  the  circuits  in  which  the  currents  flow,  and  inversely 
proportional  to  the  total  resistances  of  these  circuits  ;  that  is,  if  C 
represents  the  current  in  amperes,  E  the  E.M.F.  in  volts,  and 
R  the  resistance  of  the  circuit  in  ohms,  then  Ohm's  law  as 
applied  to  the  complete  circuit  is : 

n-    K      ,       .  electromotive  force  ^-i  x 

C  —      ;  that  is,  current  =  --        — : —  —  •          (,  I2- 

-R  resistance 


GEORG  SIMON  OHM  (1787-1854) 

German  physicist  and  discoverer  of  the  famous  law  in  physics 
which  bears  his  name.  He  was  born  and  educated  at  Erlangen. 
It  was  in  1826,  while  he  was  teaching  mathematics  at  a  gym- 
nasium in  Cologne,  that  he  published  his  famous  paper  on  the 
experimental  proof  of  his  law.  At  the"  time  of  his  death  he  was 
professor  of  experimental  physics  in  the  university  at  Munich 


ELECTROMOTIVE  FORCE  AND  RESISTANCE     253 

As  applied  to  any  portion  of  an  electrical  circuit,  Ohm's 

law  is 

_,      PI)     .,    ,  .  potential  difference 

C= ;  that  is,  current  —  —  —  ,       (2) 

r  resistance 

where  P.D.  represents  the  difference  of  potential  in  volts  be- 
tween any  two  points  in  the  circuit,  and  r  the  resistance  in 
ohms  of  the  conductor  connecting  these  two  points.  This  is 
one  of  the  most  important  laws  in  physics. 

Both  of  the  above  statements  of  Ohm's  law  are  included  in 

the  equation : 

volts  .0 

amperes  =  —     — .  (3) 

ohms 

320.  Internal  resistance  of  a  galvanic  cell.   Let  the  zinc  and 

copper  plates  of  a  simple  galvanic  cell  be  connected  to  an  ammeter,  and 
the  distance  between  the  plates  then  increased.  The  deflection  of  the 
needle  will  be  found  to  decrease;  or,  if  the  amount  of  immersion  is 
decreased,  the  current  also  will  decrease. 

Now,  since  the  E.M.F.  of  a  cell  was  shown  in  §  315  to  be 
wholly  independent  of  the  area  of  the  plates  immersed  or  of 
the  distance  between  them,  it  will  be  seen  from  Ohm's  law 
that  the  change  in  the  current  in  these  cases  must  be  due  to 
some  change  in  the  total  resistance  of  the  Circuit.  Since  the 
wire  which  constitutes  the  outside  portion  of  the  circuit  has 
remained  the  same,  we  must  conclude  that  the  liquid  within 
the  cell,  as  well  as  the  external  wire,  offers  resistance  to  the  pas- 
sage of  the  current.  This  internal  resistance  of  the  liquid  is 
directly  proportional  to  the  distance  between  the  plates,  and 
inversely  proportional  to  the  area  of  the  immersed  portion  of 
the  plates.  If,  then,  we  represent  the  external  resistance  of  the 
circuit  of  a  galvanic  cell  by  Jtc  and  the  internal  by  R#  Ohm's 
law  as  applied  to  the  entire  circuit  takes  the  form 

C=    -2-  (4) 


254  ELECTRICITY  IN  MOTION 

Thus,  if  a  simple  cell  has  an  internal  resistance  of  2  ohms  and  an 
E.M.F.  of  1  volt,  the  current  which  will  flow  through  the  circuit  when 
its  terminals  are  connected  by  9.35  ft.  of  No.  30  copper  wire  (1  ohm)  is 

=  .33  ampere. 

321.  Measurement  of  internal  resistance.    A  simple  and  direct  method 
of  finding  a  length  of  wire  which  has  a  resistance  equivalent  to  the 
internal  resistance  of  a  cell  is  to  connect  the  cell  first  to  an  ammeter 
or  any  galvanometer  of  negligible  resistance*  and  then  to  introduce 
enough  German-silver  wire  into  the  circuit  to  reduce  the  galvanometer 
reading  to  half  its  original  value.    The  internal  resistance  of  the  cell  is 
then  equal  to  that  of  the  German-silver  wire.    Why?    A  still  easier 
method  in  case  both  an  ammeter  and  a  voltmeter  are  available  is  to 
divide  the  E.1\£.F.  of  the  cell  as  given  by  the  voltmeter  by  the  current 
which  the  cell  is  able  to  send  through  the  ammeter  when  connected 
directly  to  its  terminals ;  for  in  this  case  Rc  of  equation  (4)  is  0 ;  there- 

E 

fore  7tt-  =  —  •    This  gives  the  internal  resistance  directly  in  ohms. 
C 

322.  Measurement  of  any  resistance  by  ammeter- voltmeter 
method.    The  simplest  way  of  measuring  the  resistance  of  a 
wire,  or  in  general  of  any  conductor,  is  to  connect  it  into  the 
circuit  of  a  galvanic  cell  in  the  manner 

shown  in  Fig.  260.  The  ammeter  A  is 
inserted  to  measure  the  current,  and  the 
voltmeter  V  to  measure  the  P.D.  between 
the  ends  a  and  b  of  the  wire  r,  the  resist- 
ance  of  which  is  sought.  The  resistance  FIG.  260.  Ammeter- 

of  r  in  ohms  is  obtained  at  once  from  the      voltmeter  method  of 
-,        ,^  ,.  .,,    jn  measuring  resistance 

ammeter  and  voltmeter  readings  with  the 

P.D.  P  I). 

aid  of  the  law  C-—   —•>  from  which  it  follows  that  r  =  -   — • 

r  C 

Thus,  if  the  voltmeter  indicates  a  P.D.  of  .4  volt  and  the  arnme- 

4 

ter  a  current  of  .5  ampere,  the  resistance  of  r  is  '—  —  .8  ohm.t 

.o 

*  A  lecture-table  ammeter  is  best,  but  see  note  on  page  249. 
fine  Wheatstone's  bridge  method  of  measuring  resistance  is  recommended 
for  laboratory  study.   See,  for  example,  Experiment  33  of  the  authors'  manual. 


ELECTROMOTIVE  FORCE  AND  RESISTANCE    255 

323.  Joint  resistance  of  conductors  connected  in  series  and  in 
parallel.    When  resistances  are  connected  as  in  Fig.  261,  so 

that  the  same  current  flows 

T        ,.    ,-,  -  lOtim     3Ohms       60tims      lOhm  . 

through    eacn    01    tnem   in      atfTrriroT$~ttinr^^ 

succession,  they  are  said  to 


v.  x 


be  connected  in  series.    The  FlG>  261.  Series  connections 

total  resistance  of  a  number 

of  conductors  so  connected  is  the  sum  of  the  several  resist- 
ances. Thus,  in  the  case  shown  in  the  figure,  the  total 
resistance  between  a  and  b  is  10  ohms. 

When    n    exactly    similar    conductors    are    joined    in    the 
manner    shown   in   Fig.   262,  that   is,   in  parallel,   the   total 
resistance  between  a  and  b  is  1/n  of  the  resistance  of  one  of 
them  ;  for,  obviously,  with  a  given  P.D. 
between  the  points  a  and  5,  four  conduc- 
tors will  carry  four  times  as  much  cur- 
rent as  one,  and  n  conductors  will  carry 
n  times  as  much  current  as  one.    There- 

fore  the   resistance,   which    is    inversely 

,    J      FIG.  262.   Parallel  con- 
proportional  to  the  carrying  capacity  (see  nections 

§  31  9),  is  l/n  as  much  as  that  of  one. 

Even  if  the  resistances  are  not  all  alike,  since  the  total 
current  C  between  a  and  b  is  obviously  the  sj»im  of  the  currents 
in  the  branches,  that  is,  since  C  =  Cl  +  <72  +  C8  +  C#  we  have  at 

once  the  joint  resistance  R  is  given  by  —  =  —  -(-  —  +  —  +  —  • 

H       Al       Mz       Jt8       JBj 

324.  Shunts.    A  wire   connected  in  parallel  with  another 
wire  is  said  to  be  a  shunt  to  that  wire. 
Thus  the  conductor  X  (Fig.  263)  is  said  / 


to   be   shunted  across    the    resistance  R. 

,.  .  FIG.  263.    A  shunt 

Under  such  conditions  the  currents  car- 

ried by  R  and  X  will  be  inversely  proportional  to  their  re- 
sistances, so  that,  if  X  is  1  ohm  and  12  10  ohms,  R  will  carry 
JL  as  much  current  as  X,  or  -1-  of  the  whole  current. 


256  ELECTEICITY  IK  MOTION 

QUESTIONS  AND  PROBLEMS 

1.  Two  wires  of  the  same  material  but  of  diameters  in  the  ratio 

1  to  2  are  connected  in  series  in  the  same  electrical  circuit.    The  fall  of 
potential  in  the  smaller  wire  is  2  volts  per  foot  of  length.    What  is  it  in 
the  larger  wire  ? 

2.  If  the  potential  difference  between  the  terminals  of  a  cell  on  open 
circuit  is  to  be  measured  by  means  of  a  galvanometer,  why  must  the 
galvanometer  have  a  high  resistance? 

3.  How  long  a  piece   of   No.  30  copper  wire  will  have  the  same 
resistance  as  a  meter  of  No.  30  German-silver  wire? 

4.  The  resistance  of  a  certain  piece  of  German-silver  wire  is  1  ohm. 
What  will  be  the  resistance  of  another  piece  of  the  same  length  but  of 
twice  the  diameter? 

5.  How  much  current  will  flow  between  two  points  whose  P.D.  is 

2  volts,  if  they  are  connected  by  a  wire  having  a  resistance  of  12  ohms? 

6.  What  P.D.  exists  between  the  ends  of  a  wire  whose  resistance  is 
100  ohms,  when  the  wire  is  carrying  a  current  of  .3  ampere  ? 

7.  If  a  voltmeter  attached  across  the  terminals  of  an  incandescent 
lamp  shows  a  P.D.  of  110  volts,  while  an  ammeter  connected  in  series 
with  the  lamp  (see  Fig.  260)  indicates  a  current  of  .5  ampere,  what  is 

x/^ithe  resistance  of  the  incandescent  filament  ?. 

yv"v  8.  Ten  pieces  of  wire,  each  having  a  resistance  of  5  ohms,  are  con- 
nected in  parallel  (see  Fig.  262).  If  the  junction  a  is  connected  to  one 
terminal  of  a  Daniell  cell  and  b  to  the  other,  what  is  the  total  current 
which  will  flow  through  the  circuit  when  the  E.M.F.  of  the  cell  is  1  volt 
and  its  resistance  2  ohms? 

9.  A  voltmeter  which  has  a  resistance  of  2000  ohms  is  shunted 
across  the  terminals  A  and  B  of  a  wire  which  has  a  resistance  of  1  ohm. 
What  fraction  of  the  total  current  flowing  from  A  to  B  will  be  carried 
by  the  voltmeter  ? 

10.  In  a  given  circuit  the  P.D.  across  the  terminals  of  a  resistance 
of  19  ohms  is  found  to  be  3  volts.  What  is  the  P.D.  across  the  termi- 
nals of  a  3-ohm  wire  in  the  same  circuit  ? 

PRIMARY  CELLS 

325.  Study  of  the  action  of  a  simple  cell.  If  the  simple  cell 
already  described,  that  is,  zinc  and  copper  strips  in  dilute  sulphuric  acid, 
is  carefully  observed,  it  will  be  seen  that,  so  long  as  the  plates  are  not 
connected  by  a  conductor,  fine  bubbles  of  gas  are  slowly  formed  at  the 
zinc  plate,  but  none  at  the  copper  plate.  As  soon,  however,  as  the  two 
strips  are  put  into  metallic  connection,  bubbles  appear  in  great  numbers 


PRIMARY  CELLS 


257 


FIG.  264.  Chemical 
actions  in  the  vol- 
taic cell 


about  the  copper  plate  (Fig.  264),  and  at  the  same  time  a  current  mani- 
fests itself  in  the  connecting  wire.  These  are  bubbles  of  hydrogen. 
Their  appearance  on  the  zinc  may  be  prevented 
either  by  using  a  plate  of  chemically  pure  zinc  or 
by  amalgamating  impure  zinc ;  that  is,  by  coating 
it  over  with  a  thin  film  of  mercury.  But  the  bub- 
bles on  the  copper  cannot  be  thus  disposed  of. 
They  are  an  invariable  accompaniment  of  the  cur- 
rent in  the  circuit.  If  the  current  is  allowed  to 
run  for  a  considerable  time,  it  will  be  found  that 
the  zinc  wastes  away,  even  though  it  has  been 
amalgamated,  but  that  the  copper  plate  does  not 
undergo  any  change. 

We  learn,  therefore,  that  the  electrical 
current  in  the  simple  cell  is  accompanied 
with  the  eating  up  of  the  zinc  plate  by  the  liquid,  and  by  the 
evolution  of  hydrogen  bubbles  at  the  copper  plate.  In  every 
type  of  galvanic  cell  actions  similar  to  these  two  are  always 
found  ;  that  is,  one  of  the  plates  is  always  eaten  up,  and  upon  the 
other  plate  some  element  is  deposited.  The  plate  which  is  eaten 
is  always  the  one  which  is  found  to  be  negatively  charged 
when  tested  as  in  §  310,  so  that  in  all  galvanic  cells,  when 
the  terminals  are  connected  through  a  wire,  the  negative 
electricity  flows  through  this  wire  from  the  eaten  plate  to  the 
uneaten  plate.  This  means,  in  accordance  with  the  convention 
mentioned  in  §  300,  that  the  direction  of  the  current 
through  the  external  circuit  is  always  from  the 
uneaten  to  the  eaten  plate. 

326.  Local  action  and  amalgamation.  The  cause 
of  the  appearance  of  the  hydrogen  bubbles  at  the 
surface  of  impure  zinc  when  dipped  in  dilute  sul- 
phuric acid  is  that  little  electrical  circuits  are  set 

.  .        .  FIG. 265. 

up  between  the  zinc  and  the  small  impurities  in     Local  actjon 

it,  —  carbon  or  iron  particles,  —  in   the   manner 

indicated  in  Fig.  265.    If  the  zinc  is  pure,  these  little  local 

currents  cannot,  of  course,  be  set  up,  and  consequently  no 


258 


ELECTEICITY  IN  MOTION 


hydrogen  bubbles  appear.  Amalgamation  stops  this  so-called 
local  action,  because  the  mercury  dissolves  the  zinc,  while  it 
does  not  dissolve  the  carbon,  iron,  or  other  impurities.  The 
zinc-mercury  amalgam  formed  is  a  homogeneous  substance 
which  spreads  over  the  whole  surface  and  covers  up  the 
impurities.  It  is  important,  therefore,  to  amalgamate  the  zinc 
in  a  battery  in  order  to  prevent  the  consumption  of  the  zinc 
when  the  cell  is  on  open  circuit.  The  zinc  is  under  all  circum- 
stances eaten  up  when  the  current  is  flowing,  amalgamation 
serving  only  to  prevent  its  consumption  when,  the  circuit 
is  open. 

327.  Theory  of  the  action  of  a  simple  cell.  A  simple  cell 
may  be  made  of  any  two  dissimilar  metals  immersed  in  a  solu- 
tion of  any  acid  or  salt.  For  simplicity  let  us  examine  the 
action  of  a  cell  composed  of  plates  of  zinc 
and  copper  immersed  in  a  dilute  solu- 
tion of  hydrochloric  acid.  The  chemical 
formula  for  hydrochloric  acid  is  HC1. 
This  means  that  each  molecule  of  the 
acid  consists  of  one  atom  of  hydrogen 
combined  with  one  atom  of  chlorine. 

In  accordance  with  the  theory  now  in     FIG.  266.  Showing  disso- 

-i       .    .   ,  -11         '   , 

vogue  among  physicists  and  chemises, 
.when  hydrochloric  acid  is  mixed  with 
water  so  as  to  form  a  dilute  solution,  the  HC1  molecules  split 
up  into  two  electrically  charged  parts,  called  ions,  the  hydro- 
gen ion  carrying  a  positive  charge  and  the  chlorine  ion  an 
equal  negative  charge  (Fig.  266).  This  phenomenon  is  known 
as  dissociation.  The  solution  as  a  whole  is  neutral  ;  that  is,  it 
is  uncharged,  because  it  contains  just  as  many  positive  as 
negative  ions. 

When  a  zinc  plate  is  placed  in  such  a  solution  the  acid 
attacks  it  and  pulls  zinc  atoms  into  solution.  Now  whenever 
a  metal  dissolves  in  an  acid,  its  atoms,  for  some  unknown 


elation    of    hydrochloric 

ju  w<uer 


PRIMARY  CELLS  259 

reason,  go  into  solution  bearing  little  positive  charges.  The 
corresponding  negative  charges  must  be  left  on  the  zinc  plate  in 
precisely  the  same  way  in  which  a  negative  charge  is  left  on 
silk  when  positive  electrification  is  produced  on  a  glass  rod 
by  rubbing  it  with  the  silk.  It  is  in  this  way,  then,  that  we 
account  for  the  negative  charge  which  we  found  upon  the 
zinc  plate  in  the  experiment  which  was  performed  in  con- 
nection with  §  310. 

The  passage  of  positively  charged  zinc  ions  into  solution 
gives  a  positive  charge  to  the  solution  about  the  zinc  plate, 
so  that  the  hydrogen  ions  tend  to  be  repelled  toward  the  cop- 
per plate.  When  these  repelled  hydrogen  ions  reach  the  copper 
plate,  some  of  them  give  up  their  charges  to  it  and  then  collect 
as  bubbles  of  hydrogen  gas.  It  is  in  this  way  that  we  account 
for  the  positive  charge  which  we  found  on  the  copper  plate  in 
the  experiment  of  §  310. 

If  the  zinc  and  copper  plates  are  not  connected  by  an  out- 
side conductor,  this  passage  of  positively  charged  zinc  ions 
into  solution  continues  but  a  very  short  time,  for  the  zinc  soon 
becomes  so  strongly  charged  negatively  that  it  pulls  back  on 
the  -h  zinc  ions  with  as  much  force  as  the  acid  is  pulling  them 
into  solution.  In  precisely  the  same  way  the  copper  plate  soon 
ceases  to  take  up  any  more  positive  electricity  from  the  hydro- 
gen ions,  since  it  soon  acquires  a  large  enough  -+-  charge  to 
repel  them  from  itself  with  a  force  equal  to  that  with  which 
they  are  being  driven  out  of  solution  by  the  positively  charged 
zinc  ions.  It  is  in  this  way  that  we  account  for  the  fact  that 
on  open  circuit  no  chemical  action  goes  on  in  the  simple  gal- 
vanic cell,  the  zinc  and  copper  plates  simply  becoming  charged 
to  a  definite  difference  of  potential  which  is  called  the  E.M.F. 
of  the  cell. 

When,  however,  the  copper  and  zinc  plates  are  connected 
by  a  wire,  a  current  at  once  flows  from  the  copper  to  the  zinc 
and  the  plates  thus  begin  to  lose  their  charges.  This  allows 


262  ELECTRICITY  IN  MOTION 

copper  sulphate  solution  of  positive  copper  ions  and  negative 
SO4  ions.  Now  the  zinc  of  the  zinc  plate  goes  into  solution 
in  the  zinc  sulphate  in  precisely  the  same  way  that  it  goes 
into  solution  in  the  hydrochloric  acid  of  the  simple  cell  de- 
scribed in  §  327.  This  gives  a  positive  charge  to  the  solution 
about  the  zinc  plate,  and  causes  a  movement  of  the  positive 
ions  between  the  two  plates  from  the  zinc  toward  the  copper, 
and  of  negative  ions  in  the  opposite  direction,  both  the  Zn  and 
the  SO4  ions  being  able  to  pass  through  the  porous  cup.  Since 
the  positive  ions  about  the  copper  plate  consist  of  atoms  of 
copper,  it  will  be  seen  that  the  material  which  is  driven  out 
of  solution  at  the  copper  plate,  instead  of  being  hydrogen,  as 
in  the  simple  cell,  is  metallic  copper.  Since,  then,  the  element 
which  is  deposited  on  the  copper  plate  is  the  same  as  that  of 
which  it  already  consists,  it  is  clear  that  neither  the  E.M.F. 
nor  the  resistance  of  the  cell  can  be  changed  because  of  this 
deposit ;  that  is,  the  cause  of  the  polarization  of  the  simple  cell 
has  been  removed. 

The  great  advantage  of  the  Daniell  cell  lies  in  the  relatively 
high  degree  of  constancy  in  its  E.M.F.  (1.08  volts).  It  has  a 
comparatively  high  internal  resistance  (one  to  six  ohms)  and 
is  therefore  incapable  of  producing  very  large  currents,  about 
one  ampere  at  most/  It  will  furnish  a  very  constant  current, 
however,  for  a  great  length  of  time,  in  fact,  until  all  of  the 
copper  is  driven  out  of  the  copper  sulphate  solution.  In  order 
to  keep  a  constant  supply  of  the  copper  ions  in  the  solution, 
copper  sulphate  crystals  are  kept  in  the  compartment  S  of  the 
cell  of  Fig.  267,  or  in  the  bottom  of  the  gravity  cell.  These 
dissolve  as  fast  as  the  solution  loses  its  strength  through  the 
deposition  of  copper  011  the  copper  plate. 

The  Daniell  is  a  so-called  "closed-circuit"  cell;  that  is,  its 
circuit  should  be  left  closed  (through  a  resistance  of  thirty  or 
forty  ohms)  whenever  the  cell  is  not  in  use.  If  it  is  left  on 
open  circuit,  the  copper  sulphate  diffuses  through  the  porous 


PRIMARY  CELLS 


263 


FIG.  269.  The  Western 
normal  cell 


cup  and  a  brownish  muddy  deposit  of  copper  or  copper  oxide 
is  formed  upon  the  zinc.  Pure  copper  is  also  deposited  in  the 
pores  of  the  porous  cup.  Both  of  these  actions  damage  the 
cell.  When  the  circuit  is  closed,  however,  since  the  electrical 
forces  always  keep  the  copper  ions  moving  toward  the  copper 
plate,  these  damaging  effects  are  to  a 
large  extent  avoided. 

331.  The  Weston  normal  cell ;  the  volt. 
This  cell  consists  of  a  positive  electrode 
of  mercury  in  a  paste  of  mercurous  sul- 
phate,  and  a  negative  electrode  of  cad- 
mium amalgam  in  a  saturated  solution  of 
cadmium   sulphate    (Fig.  269).    It   is  so 
easily  and  exactly  reproducible  and  has 
an    E.M.F.    of    such    extraordinary    con- 
stancy that  it  has  been  taken  by  international  agreement  as  the 
standard  in  terms  of  which  all  E.M.F.'s  and  P.D.'s  are  rated. 

Thus  the  E.M.F  of  a  Weston  normal  cell  at  20°  O.  is  taken  as 
1.0183  volts.  The  legal  definition  of  the  volt  is  then  an  electrical 
pressure  equal  to  1  Q11  g  3  of  that  produced  by  a  Weston  normal  cell. 

332.  The  Leclanch6  cell.    The  Leclanche"  cell  (Fig.  270)  consists  of  a 
zinc  rod  in  a  solution  of  ammonium  chloride  (150  grams  to  a  liter  of 
water),  and  a  carbon  plate  placed  inside  of  a  porcAis 

cup  which  is  packed  full  of  manganese  dioxide  and 
powdered  graphite  or  carbon.  As  in  the  simple 
cell,  the  zinc  dissolves  in  the  liquid  and  hydrogen 
is  liberated  at  the  carbon,  or  positive,  plate.  Here 
it  is  slowly  attacked  by  the  manganese  dioxide. 
This  chemical  action  is,  however,  not  quick  enough 
to  prevent  rapid  polarization  when  large  currents 
are  taken  from  the  cell.  The  cell  slowly  recovers 
when  allowed  to  stand  for  a  while  on  open  cir- 
cuit. The  E.M.F.  of  a  Leclanche"  cell  is  about  1.5 
volts,  and  its  initial  internal  resistance  is  somewhat  less  than  an 
ohm.  It  therefore  furnishes  a  momentary  current  of  from  one  to 
three  amperes. 


FIG.  270 
The  Leclanclie"  cell 


264  ELECTRICITY  IN  MOTION 

The  immense  advantage  of  this  type  of  cell  lies  in  the  fact  that  the 
zinc  is  not  at  all  eaten  by  the  ammonium  chloride  when  the  circuit  is 
open,  and  that  therefore,  unlike  the  Daniell  cell,  it  can  be  left  for  an 
indefinite  time  on  open  circuit  without  deterioration.  Leclanch6  cells 
are  used  almost  exclusively  where  momentary  currents  only  are  needed, 
as,  for  example,  on  doorbell  circuits.  The  cell  requires  no  attention  for 
years  at  a  time,  other  than  the  occasional  addition  of  water  to  replace 
loss  by  evaporation,  and  the  occasional  addition  of  ammonium  chloride 
(NH4C1)  to  keep  positive  NH4  and  negative  Cl  ions  in  the  solution. 

333.  The  dry  cell.    The  dry  cell  is  only  a  modified  form  of  the 
LeclanchS  cell.  It  is  not  really  dry,  since  the  zinc  and  carbon  plates  are 
embedded  in  moist  paste  which  consists  usually  of  one  part  of  crystals  of 
ammonium  chloride,  three  parts  of  plaster  of  Paris,  one  part  of  zinc  oxide, 
one  part  of  zinc  chloride,  and  two  parts  of  water.    The  plaster  of  Paris  is 
used  to  give  the  paste  rigidity.    As  in  the  Leclanche  cell,  it  is  the  action 
Of  the  ammonium  chloride  upon  the  zinc  which  produces  the  current. 

334.  Combinations  of  cells.    There  are  two  ways  in  which 
cells  may  be  combined :  first,  in  series ;  and  second,  in  parallel. 
When  they  are  connected  in  series  the  zinc  of  one  cell  is  joined 
to  the  copper  of  the  second,  the  zinc  of  the 

second  to  the  copper  of  the  third,  etc.,  the 
copper  of  the  first  and  the  zinc  of  the  last 
being  joined  to  the  ends  of  the  external  re- 
sistance (see  Fig.  271).  The  E.M.F.  of  such  FlG- 271-  Cells  con- 

,  .      , .        .     ,,  £  ,,      ^  ,  r  T^  ,       -         nected  in  series 

a  combination  is  the  sum  of  the  E.M.E .  s  of 

the  single  cells.  The  internal  resistance  of  the  combination  is 
also  the  sum  of  the  internal  resistances  of  the  single  cells. 
Hence,  if  the  external  resistances  are  very  small,  the  current 
furnished  by  the  combination  will  be  no  larger  than  that 
furnished  by  a  single  cell,  since  the  total  resistance  of  the  cir- 
cuit has  been  increased  in  the  same  ratio  as  the  total  E.M.F. 
But  if  the  external  resistance  is  large,  the  current  produced 
by  the  combination  will  be  very  much  greater  than  that  pro- 
duced by  a  single  cell.  Just  how  much  greater  can  always  be 
determined  by  applying  Ohm's  law,  for  if  there  are  n  cells  in 
series,  and  E  is  the  E.M.F.  of  each  cell,  the  total  E.M.F. 


PRIMARY  CEfiLS 


265 


of  the  circuit  is  nE.   Hfence,  if  R-*  is  the  external  resistance  and 
Ri  the  internal  resistance  of  a  single  cell,  then  Ohm's  law  gives 


nE 


Re  +  nRt 

If  the  n  cells  are  connected  in  parallel, 
that  is,  if  all  the  coppers  are  connected 
together  and  all  the  zincs,  as  in  Fig.  273, 
the  E.M.F.  of,  the  combination  is  only  the 
E.M.F.  of  a  single  cell,  while  the  internal 
resistance  is  \/n  of  that  of  a  single  cell, 
since  connecting  the  cells  in  this  way  is 
simply  equivalent  to  multi- 
plying the  area  of  the  plates 
n  times.  The  current  fur- 
nished by  such  a  combina- 


FIG.  272.   Water  anal- 
ogy of  cells  in  series 


tion  will  be  given  by  the  formula 


C  = 

FIG.  273.  Cells 
in  parallel 

If,  therefore,  Re  is  negligibly  small,  as  in  the  case  of  a  heavy 
copper  wire,  the  current  flowing  through  ifc  will  be  n  times  as 
great  as  that  which  could  be  made 
to  flow  through  it  by  a  single  cell. 
Figs.  272  and  274  show  by  means  of 
the  water  analogy  why  the  E.M.F.  of 
cells  in  series  is  the  sum  of  the  several 
E.M.F.'s  and  why  the  J&M.F.  of  cells 
in  parallel  is  no  greater  than  that  of  a 
single  cell.  These  considerations- show 
that  the  rules  which  should  govern  titie 
combination  of  ceBs  are  as  follows :  Connect  in  series  when  Re 
is  large  in  comparismi  with  7£* ;  connect  in  parallel  when  Ri  is 
large  in  comparison  with  Re. 


FIG.  274.    Water  analogy 
of  cells  in  parallel* 


266  ELECTRICITY  IN  MOTION 

QUESTIONS  AND  PROBLEMS 

1.  A  Daniell  cell  is  found  to  send  a  current  of  .5  ampere  through  an 
ammeter  of  negligible  resistance.    What  is  its  internal  resistance? 

2.  Why  is  a  Leclanche  cell  better  than  a  Daniell  cell  for  ringing 
doorbells? 

3.  If  the  internal  resistance  of  a  Daniell  cell  of  the  gravity  type  is 
4  ohms,  and  its  E.M.F.  1.08  volts,  how  much  current  will  40  cells  in 
series  send  through  a  telegraph  line  having  a  resistance  of  500  ohms  ? 
What  current  will  one  such  cell  send  through  the  same  circuit?    What 
current  will  40  cells  joined  in  parallel  send  through  the  same  circuit? 

4.  What  current  will  the  40  cells  in  parallel  send  through  an  am- 
meter which  has  a  resistance  of  .1  ohm?    What  current  would  the 
40  cells  in  series  send  through  the  same  ammeter  ?    What  current  would 
a  single  cell  send  through  the  same  ammeter  ? 

5.  Can  you  prove  from  a  consideration  of  Ohm's  law  that  when 
wires  of  different  resistances  are  inserted  in  series  in  a  circuit,  the  P.D.'s 
between  the  ends  of  the  various  wires  are  proportional  to  the  resistances 
of  these  wires  ? 

6.  Under  what  conditions  will  a  small  cell  give  practically  the  same 
current  as  a  large  one  of  the  same  type  ? 

7.  Why  must  a  galvanometer  which  is  to  be  used  for  measuring 
voltages  have  a  high  resistance  ? 

8.  Why  is  it  desirable  that  a  galvanometer  which  is  to  be  used  for 
measuring  currents  have  as  small  a  resistance  as  possible? 

9.  A  50-volt  lamp,  resistance  110  ohms,  and  a  110-volt  lamp,  re- 
sistance   220  ohms,    are   connected   in  parallel.     What  is  their  joint 
resistance  ? 

10.   With  the  aid  of  Figs.  272  and  274  discuss  the  water  analogies 
of  the  rules  on  the  bottom  of  page  265. 


CHAPTER    XIV 

CHEMICAL,  MAGNETIC,  AND  HEATING  EFFECTS  OF  THE 
ELECTRIC  CURRENT 

CHEMICAL  EFFECTS  ;  ELECTROLYSIS  * 

335.  Electrolysis.  Let  two  platinum  electrodes  be  dipped  into  a 
solution  of  dilute  sulphuric  acid,  and  let  the  terminals  of  a  battery 
producing  an  E.M.F.  of  10  volts  or  more  be  applied  to  these  electrodes. 
Oxygen  gas  is  found  to  be  given  off  at  the  electrode  at  which  the  cur- 
rent enters  the  solution,  called  the  anode,  while  hydrogen  is  given  off 
at  the  electrode  at  which  the  current  leaves  the  solution,  called  the 
cathode.  These  gases  may  be  collected  in  test 
tubes  in  the  manner  shown  in  Fig.  275. 

The  modern  theory  of  this  phenom- 
enon is  as  follows :  Sulphuric  acid 
(H2SO4),  when  it  dissolves  in  water, 
breaks  up  into  positively  charged  hy- 
drogen ions  and  negatively  charged  FlG-  275-  The  electrolysis 

c<r\    •  A  j.  •     i  £   i  i  °f  water 

o(J4  ions.   As  soon  as  an  electrical  neld 

is  established  in  the  solution  by  connecting  the  electrodes  to 
the  positive  and  negative  terminals  of  a  battery,  the  hydrogen 
ions  begin  to  migrate  toward  the  cathode,  and  there,  after 
giving  up  their  charges,  unite  to  form  molecules  of  hydrogen 
gas.  On  the  other  hand,  the  negative  SO4  ions  migrate  to  the 
positive  electrode  (that  is,  the  anode),  where  they  give  up 
their  charges  to  it,  and  then  act  upon  the  water  (H2O),  thus 
forming  H2SO4  and  liberating  oxygen. 

*  This  subject  should  he  accompanied  or  followed  by  a  laboi'atory  experiment 
on  electrolysis  and  the  principle  of  the  storage  battery.  See,  for  example,  Experi- 
ment 35  of  the  authors'  manual. 

267 


268         EFFECTS  OF  THE  ELECTEIC  CURRENT 

If  the  volumes  of  hydrogen  and  of  oxygen  are  measured, 
the  hydrogen  is  found  to  occupy  in  every  case  just  twice  the 
volume  occupied  by  the  oxygen.  This  is,  indeed,  one  of  the 
reasons  for  believing  that  water  consists  of  two  atoms  of 
hydrogen  and  one  of  oxygen. 

336.  Electroplating.    If  the  solution,  instead  of  being  sul- 
phuric acid,  had  been  one  of  copper  sulphate  (CuSO4),  the 
results  would  have  been  precisely  the  same  in  every  respect, 
except  that,  since  the  hydrogen  ions  in  the  solution  are  now 
replaced  by  copper  ions,  the  substance  deposited  on  the  cathode 
is  pure   copper  instead   of  hydrogen.    This  is  the  principle 
involved  in  electroplating  of  all  kinds.    In  commercial  work 
the  positive  plate,  that  is,  the 

plate  at  which  the  current  en- 
ters the  bath,  is  always  made 
from  the  same  metal  as  that 

which  is  to  be  deposited  from 

,1          -,    ,  •        r      •    j_i  •  ^i  FIG.  276.    Electroplating  bath 

the  solution,  tor  in  this  case  the 

SO4  or  other  negative  ions  dissolve  this  plate  as  fast  as  the  metal 
ions  are  deposited  upon  the  other.  The  strength  of  the  solution, 
therefore,  remains  unchanged.  In  effect,  the  metal  is  simply 
taken  from  one  plate  and  deposited  011  the  other.  Fig.  276  rep- 
resents a  silver-plating  bath.  The  bars  joined  to  the  anode  A 
are  of  pure  silver.  The  spoons  to  be  plated  are  connected  to 
the  cathode  K.  The  solution  consists  of  500  grams  of  potassium 
cyanide  and  250  grams  of  silver  cyanide  in  10  liters  of  water. 

337.  Electrotyping.   In  the  process  of  electrotyping  the  page 
is  first  set  up  in  the  form  of  common  type.    A  mold  is  then 
taken  in  wax  or  gutta-percha.    This  mold  is  then  coated  with 
powdered  graphite  to  render  it  a  conductor,  after  which  it  is 
ready  to  be  suspended  as  the  cathode  in  a  copper-plating  bath, 
the  anode  being  a  plate  of  pure  copper  and  the  liquid  a  solu- 
tion of  copper  sulphate.    When  a  sheet  of  copper  as  thick  as 
a  visiting  card  has  been  deposited  011  the  mold,  the  latter  is 


CHEMICAL  EFFECTS;  ELECTKOLYSIS  269 

removed  and  the  wax  replaced  by  a  type-metal  backing,  to 
give  rigidity  to  the  copper  films.  From  such  a  plate  as  many 
as  a  hundred  thousand  impressions  may  be  made.  Practically 
all  books  which  run  through  large  editions  are  printed  from 
such  electrotypes. 

338.  Refining  of  metals.    If  the  solution  consists  of  pure  copper  sul- 
phate, it  is  not  necessary  that  the  anode  be  of  chemically  pure  copper 
in  order  to  obtain  a  pure  copper  deposit  on  the  cathode.    Electrolytic 
copper,  which  is  the  purest  copper  on  the  market,  is  obtained  as  follows  : 
The  unrefined  copper  is  used  as  an  anode.    As  it  is  eaten  up  the  impuri- 
ties contained  in  it  fall  as  a  residue  to  the  bottom  of  the  tank  and  pure 
copper  is  deposited  on  the  cathode  by  the  current.    This  method  is  also 
extensively  used  in  the  refining  of  metals  other  than  copper. 

339.  Legal  units  of  current  and  quantity.    In  1834  Faraday 
found  that  a  given  current  of  electricity  flowing  for  a  given 
time  always  deposits  the  same  amount  of  a  given  element  from 
a  solution,  whatever  be  the  nature  of  the  solution  which  con- 
tains the  element.    For  example,  one  ampere  always  deposits 
in  an  hour  4.025  grams  of  silver,  whether  the  electrolyte  is 
silver  nitrate,  silver  cyanide,  or  any  other  silver  compound. 
Similarly,  an  ampere  will  deposit  in  an  hour  1.181  grams  of 
copper,  1.203  grams  of  zinc,  etc.    Faraday  further  found  that 
the  amount  of  metal  deposited  in  a  given  cell  depended  solely 
011  the  product  of  the  current  strength  by  the  time  ;  that  is,  on 
the  quantity  of  electricity  which  had  passed  through  the  cell. 
These  facts  are  made  the  basis  of  the  legal  definitions  of  current 
and  quantity,  thus : 

The  unit  of  quantity,  called  the  coulomb,  is  the  quantity  of 
electricity  required  to  deposit  .001118  gram  of  silver. 

The  unit  of  current,  the  ampere,  is  the  current  which  will 
deposit  .001118  gram  of  .silver  in  one  second. 

340.  Storage  batteries.    Let  two  6  by  8  inch  lead  plates  be  screwed 
to  a  half-inch  strip  of  some  insulating  material,  as  in  Fig.  277,  and 
immersed  in  a  solution  consisting   of   one  part  of  sulphuric  acid  to 


270         EFFECTS' OF  THE  ELECTRIC  'CURRENT' 

ten  parts  of  water.  Let  a  current  from  two  storage  or  three  dry  cells  in 
series,  C,  be  sent  through  this  arrangement,  an  ammeter  A  or  any  low- 
resistance  galvanometer  being  inserted  in  the  circuit.  As  the  current 
flows,  hydrogen  bubbles  will  be  seen  to  rise  from  the  cathode  (the  plate 
at  which  the  current  leaves  the  solution),  while  the  positive  plate,  or 
anode,  will  begin  to  turn  dark  brown.  At  the  same  time  the  reading  of 
the  ammeter  will  be  found  to  decrease  rapidly.  The  brown  coating  is  a 
compound  of  lead  and  oxygen,  called  lead 
peroxide  (PbO2),  which  is  formed  by  the 
action  upon  the  plate  of  the  oxygen  which 
is  liberated  precisely  as  in  the  experiment 
on  the  electrolysis  of  water  (§  335).  Let 
now  the  batteries  be  removed  from  the 
circuit  by  opening  the  key  Kv  and  let  an  FK,  27?>  Th(J  principle  of 
electric  bell  B  be  inserted  in  their  place  tjlc  storage  battery 

by  closing  the  key  Kz.    The  bell  will  ring 

and  the  ammeter  A  will  indicate  a  current  flowing  in  a  direction  oppo- 
site to  that  of  the  original  current.  This  current  will  decrease  rapidly 
as  the  energy  which  was  stored  in  the  cell  by  the  original  current  is 
expended  in  ringing  the  bell. 

This  experiment  illustrates  the  principle  of  the  storage  lat- 
tery. Properly  speaking,  there  has  been  110  storage  of  electricity, 
but  only  a  storage  of  chemical  energy . 

Two  similar  lead  plates  have  been  changed  by  the  action  of 
the  current  into  two  dissimilar  plates,  one  of  lead  and  one  of 
lead  peroxide.  In  other  words,  an  ordinary  galvanic  cell  has 
been  formed ;  for  any  two  dissimilar  metals  in  an  electrolyte 
constitute  a  primary  galvanic  cell.  In  this  case  the  lead  per- 
oxide plate  corresponds  to  the  copper  of  an  ordinary  cell,  and 
the  lead  plate  to  the  zinc.  This  cell  tends  to  create  a  current 
opposite  in  direction  to  that  of  the  charging  current ;  that  is, 
its  E.M.F.  pushes  back  against  the  E.M.F.  of  the  charging 
cells.  It  was  for  this  reason  that  the  ammeter  reading  fell. 
When  the  charging  current  is  removed  this  cell  acts  exactly 
like  a  primary  galvanic  cell  and  furnishes  a  current  until  the 
thin  coating  of  peroxide  is  used  up.  The  only  important  differ- 
ence between  a  commercial  storage  cell  (Fig.  278)  and  the 

t 


CHEMICAL  EFFECTS;  ELECTROLYSIS  271 


FIG.  278.   Lead-plate 
storage  cell 


one  which  we  have  here  used  is  that  the  former  is  provided  in 
the  making  with  a  much  thicker  coat  of  the  "  active  material " 
(lead  peroxide  on  the  positive  plate  and 
a  porous,  spongy  lead  on  the  negative) 
than  can  be  formed  by  a  single  charg- 
ing suci  as  we  used.  This  material  is 
pressed  into  interstices  in  the  plates, 
as  shown  in  Fig.  278.  The  E.M.F.  of 
the  storage  cell  is  about  2  volts.  Since 
the  plates  are  always  very  close  to- 
gether and  may  be  given  any  desired 
size,  the  internal  resistance  is  usually 
small,  so  that  the  currents  furnished 
may  be  very  large. 

The  usual  efficiency  of  the  storage 
cell  is  about  75%;  that  is,  only  about  three  fourths  as  much 
electrical  energy  can  be  obtained  from  it  as  is  put  into  it. 

QUESTIONS  AND  PROBLEMS 

1.  The  coulomb  (§  339)  is  3  billion  times  as  large  as  the  electrostatic 
unit  of  quantity  defined  in  §  285.    How  many  electrons  pass  per  second 
past  a  given  point  on  a  lamp  filament  which  is  carrying  1  ampere  of 
current  (see  §  290)  ? 

2.  If  the  terminals  of  a  battery  are  immersed  rh  a  glass  of  acidulated 
water,  how  can  you  tell  from  the  rate  of  evolution  of  the  gases  at  the 
two  electrodes  which  is  positive  and  which  negative  ? 

3.  How  long  will  it  take  a  current  of  1  ampere  to  deposit  1  g.  of 
silver  from  a  solution  of  silver  nitrate? 

4.  If  the  same  current  used  in  Problem  3  were  led  through  a  solution 
containing  a  zinc  salt,  how  much  zinc  would  be  deposited  in  the  same  time  ? 

5.  In  calibrating  an  ammeter,  the  current  which  produces  a  certain 
deflection  is  found  to  deposit  J  g.  of  silver  in  50  min.    What  is  the 
strength  of  the  current? 

6.  Why  is  it  possible  to  get  a  much  larger  current  from  a  storage 
cell  than  from  a  Daniell  cell  ? 

7.  A  certain  storage  cell  having  an  E.M.F.  of  2  volts  is  found  to  fur- 
nish a  current  of  20  amperes  through  an  ammeter  whose  resistance  is 
.05  ohm.    Find  the  internal  resistance  of  the  cell. 


272        EFFECTS  OF  THE  ELECTRIC  CURRENT 


MAGNETIC  PROPERTIES  OF  COILS 
341.  Loop  of  wire  carrying  a  current  equivalent  to  a  magnet 

disk.  Let  a  single  loop  of  wire  be  suspended  from  a  thread  in  the 
manner  shown  in  Fig.  279,  so  that  its  ends  dip  into  two  mercury  cups. 
Then  let  the  current  from  three  or  four  dry  cells  be 
sent  through  the  loop.  The  latter  will  be  found  to 
slowly  set  itself  so  that  the  face  of  the  loop  from 
which  the  magnetic  lines  emerge,  as  given  by  the 
righMiand  rule  (see  §  312  and  also  Fig.  280),  is  to- 
ward the  north.  Let  a  bar  magnet  be  brought  near 
the  loop.  The  latter  will  be  found  to  behave  toward 
the  magnet  in  all  respects  as  though  it  were  a  flat 
magnetic  disk  whose  boundary  is  the  wire,  the  face 

which  turns  toward  the 

north  being  an  TV  pole 

and  the  other  an  £  pole. 


FIG.  279.    A   loop 

equivalent  to  a  flat 

magnetic  disk 


lines  emerge ;  south  face  is 
face  into  which  they  enter 


Tire     experiment 
shows  what  position 
a  loop  bearing  a  current  will  always 
tend  to  assume  in  a  magnetic  field. 

FIG.  280.  North  pole  of  disk    For  since  a  magnet 
is  face  from  which  magnetic     always  tends  to  set 

itself    so    that    the 

line    connecting   its 

poles  is  parallel  to  the  direction  of  the  mag- 
netic lines  of  the  field  in  which  it  is  placed, 
a  loop  must  set  itself  so  that  a  line  con- 
necting its'  magnetic  poles  is  parallel  to  the 
lines  of  the  magnetic  field,  that  is,  so  that 
the  plane  of  the  loop  is  perpendicular  to  the  FIG.  281.  Position 
field  (see  Fig.  281)  ;  or,  to  state  the  same  assumed  by  a  loop 

thing  in  Slightly  different  form,  if  a  loop  of    Carrying  a  current 
.    "  '    J  r    J      in  a  magnetic  field 

wire,  free  to  turn,  is  carrying  a  current  in  a 

magnetic  field,  the  loop  tvill  set  itself  so  as  to  include  as  many 
as  possible  of  the  lines  of  force  of  the  field. 


MAGNETIC  PKOPEKTIES  OF  COILS 


273 


342.  Helix  carrying  a  current Aequivalent  to  a  bar  magnet. 

Let  a  wire  bearing  a  current  be  wound  in  the  form  of  a  helix  and  held 
near  a  suspended  magnet,  as  in  Fig.  282.  It  will  be  found  to  act  in  every 
respect  like  a  magnet,  with  an  N  pole  at 
one  end  and  an  S  pole  at  the  other. 


This  result  might  have  been  pre- 
dicted from  the  fact  that  a  single 
loop  is  equivalent  to  a  flat-disk 
magnet.  For  when  a  series  of  such  FlG-  282-  Magnetic  effect  of  a 

IIP!  ix 

disks  is  placed  side  by  side,  as  in  the 

helix,  the  result  must  be  the  same  as  placing  a  series  of  disk 

magnets  in  a  row,  the  N  pole  of  one  being  directly  in  contact 

with  the  S  pole  of  the  next,  etc.    These  poles  would  therefore 

all  neutralize  each  other  except 

at  the  two  ends.     We  therefore 

get  a  magnetic  field  of  the  shape 

shown  in  Fig.  283,  the  direction  of 

the  arrows  representing  as  usual 

the  direction  in  which  an  N  pole 

tends  to  move. 

The  right-hand  rule  as  given 
in  §  312  is  sufficient  in  every  case  to  determine  which  is  the  N 
and  which  the  S  pole  of  a  helix ;  that  is,  from  which  end  the 
lines  of  magnetic  force  emerge  from  the  helix  and  at  which 
end  they  enter  it.  But  it  is  found  con- 
venient, in  the  consideration  of  coils, 
to  restate  the  right-hand  rule  in  a 
slightly  different  way,  thus :  If  the  coil 
is  grasped  in  the  right  hand  in  such  a 
way  that  the  fingers  point  in  the  direc- 
tion in  which  the  current  is  flowing  in 
the  wires,  the  thumb  will  point  in  the  direction  of  the  north  pole 
of  the  helix  (see  Fig.  284).  Similarly,  if  the  sign  of  the  poles 
is  known,  but  the  direction  of  the  current  unknown,  it  may 


FIG.  283.    Magnetic  field  of  helix 


FIG.  284.   Rule  for  poles  of 
helix 


274         EFFECTS  OF  THE  ELECTRIC  CURRENT 


be  determined  as  follows :  If  the  right  hand  is  placed  against 
the  coil  with  the  thumb  pointing  in  the  direction  of  the  lines  of 
force  (that  is,  toward  the  north  pole 
of  the  helix),  the  fingers  will  pass 
around  the  coil  in  the  direction  in 
which  the  current  is  flowing. 


FIG.  285.    The  bar  electro- 
magnet 


343.  The  electromagnet.  Let  a  core 

of    soft  iron   be    inserted   in  the  helix 

(Fig.  285).   The  poles  will  be  found  to  be 

enormously  stronger  than  before.    This 
j   is  because  the  core  is  magnetized  by  induction  from  the  field  of  the 

helix  in  precisely  the  same  way  in  which  it  would  be  magnetized  by 

induction  if  placed  in  the  field  of  a  perma- 
|  nent  magnet.  The  new  field  strength  about 

the  coil  is  now  the  sum  of  the  fields  due 

to  the  core  and  that  due  to  the  coil.   If  the 
J  current  is  broken,  the  core  will  at  once 

lose  the  greater  part  of  its  magnetism.    If 

the  current  is  reversed,  the  polarity  of  the 

core  will  be  reversed.    Such  a  coil  with  a 

soft-iron  core  is  called  an  electromagnet.  pIG.  286.    The  horseshoe 

electromagnet 
The  strength  of  an  electromagnet 

can  be  very  greatly  increased  by  giving  it  such  form  that 
the  magnetic  lines  can  remain  in  iron  throughout  their  entire 
length  instead  of  emerging  into  air,  as  they 
do    in    Fig.  285.    For  this  reason  electro- 
magnets are  usually  built  in  the  horseshoe 
form    and    provided    with    an    armature   A 
(Fig.  286),  through  which  a  complete  iron 
path  for  the  lines  of  force  is  established,  as       L- : 
shown  in  Fig.  287.    The  strength  of  such  a    FIG.  287.  Magnetic 
magnet  depends  chiefly  upon   the  number    circuit  of  an^lec 
of  ampere  turns  which  encircle  it,  the  expres- 
sion "  ampere  turns  "  denoting  the  product  of  the  number  of 
turns  of  wire  about  the  magnet  by  the  number  of  amperes 


MAGNETIC  PROPERTIES  OF  COILS 


275 


FIG.  288.    Construction  of  a  com- 
mercial ammeter 


flowing  in  each  turn.  Thus  a  current  of  j~  ampere  flowing 
1000  times  around  a  core  will  make  an  electromagnet  of 
precisely  the  same  strength  as 
a  current  of  1  ampere  flowing 
10  times  about  the  core^, 

344.  Commercial    ammeters 
and  voltmeters.  Fig.  288  shows 
the  construction   of  the   usual 
form   of   commercial  ammeter. 
The  coil  c  is  pivoted  on  jewel 
bearings  and  is  held  at  its  zero 
position  by  a  spiral  spring  p. 
When  a  current  flows  through 
the  instrument,  were  it  not  for 
the  spring  p  the  coil  would  turn 
through  about  120°,  or  until  its 
n  pole  came  opposite  the  S  pole 

of  the  magnet  (see  Fig.  281).  This  zero  position  of  the  coil  is 
chosen  because  it  enables  the  scale  divisions  to  be  nearly  equal. 
The  shunt  coils  r  are  of  practically  negligible  resistance. 

The  voltmeter  differs  from  the  am- 
meter only  in  that  the  coils  r  are  in 
series  with  c  and  are  of  high  resistance. 
The  same  instrument  may  have  its  range 
changed  or  may  even  be  used  inter- 
changeably as  an  ammeter  or  a  volt- 
meter by  suitably  changing  the  coils  r. 

345.  The  electric  bell.    The   electric  bell 
(Fig.  289)  is  one  of  the  simplest  applications 
of  the  electromagnet.    When  the   button  P 
(Figs.  289  and  290)  is  pressed,  the  electric 

circuit  of  the  battery  is  closed,  and  a  current  flows  in  at  A,  through  the 
coils  of  the  magnet,  over  the  closed  contact  C,  and  out  again  at  B.  But 
no  sooner  is  this  current  established  than  the  electromagnet  E  pulls 
over  the  armature  a,  and  in  so  doing  breaks  the  contact  at  C.  This  stops 


FIG.  289.    The  electric 
bell 


276        EFFECTS  OF  THE  ELECTKIC  CUBKENT 


FIG.  290.    Cross  section  of 
the  electric  push  button 


the  current  and  demagnetizes  the  magnet  E.  The  armature  is  then 
thrown  back  against  C  by  the  elasticity  of  the  spring  s  which  supports 
it.  No  sooner  is  the  contact  made  at  Cthan  the  current  again  begins  to 
flow  and  the  former  operation  is  repeated. 
Thus  the  circuit  is  automatically  made  and 
broken  at  C,  and  the  hammer  //  is  in  conse- 
quence set  into  rapid  vibration  against  the 
rim  of  the  bell. 

346.  The  telegraph.  The  electric  telegraph 
is  another  simple  application  of  the  electro- 
magnet. The  principle  is  illustrated  in  Fig.  291.  As  soon  as  the  key  A' 
at  Chicago,  for  example,  is  closed,  the  current  flows  over  the  line  to, 
we  will  say,  New  York.  There  it  passes  through  the  electromagnet  w, 
and  thence  back  to  Chicago  through  the  earth.  The  armature  I  is  held 
down  by  the  electromagnet  m  as  long  as  the  key  K  is  kept  closed.  As 
soon  as  the  circuit  is  broken  at  K  the  armature  is  pulled  up  by  the 
spring  d.  By  means  of  a  clockwork  device  the  tape  c  is  drawn  along  at 
a  uniform  rate  beneath  the  pencil  or  pen  carried  by  the  armature  I.  A 
very  short  time  of  closing  of  K  produces  a  dot  upon  the  tape,  a  longer 
time  a  dash.  As  the  Morse,  or  telegraphic,  alphabet  consists  of  certain 
combinations  of  dots  and  dashes,  any  desired  message  may  be  sent  from 
Chicago  and  recorded  in  New  York.  In  modern  practice  the  message  is 


Chicago 


FIG.  291.    Principle  of  the  telegraph 


not  ordinarily  recorded  on  a  tape,  for  operators  have  learned  to  read  mes- 
sages by  ear,  a  very  short  interval  between  two  clicks  being  interpreted 
as  a  dot,  a  longer  interval  as  a  dash. 

The  first  commercial  telegraph  line  was  built  by  S.  F.  B.  Morse 
between  Baltimore  and  Washington.  It  was  opened  on  May  24,  1844, 
with  the  now  famous  message,  "  What  hath'  God  wrought !  " 

347.  The  relay  and  sounder.  On  account  of  the  great  resistance  of 
long  lines,  the  current  which  passes  through  the  electromagnet  is  so 
weak  that  the  armature  of  this  magnet  must  be  made  very  light  in 
order  to  respond  to  the  action  of  the  current.  The  clicks  of  such  an 


SAMUEL  F.  B.  MORSE  (1791-1872) 

The  inventor  of  the  electromagnetic  recording  telegraph  and  of  the 
dot-and-dash  alphabet  known  by  his  name,  was  born  at  Charles- 
town,  Massachusetts,  graduated  at  Yale  College  in  1810,  invented 
the  commercial  telegraph  in  1832,  and  struggled  for  twelve  years 
in  great  poverty  to  perfect  it  and  secure  its  proper  presentation 
to  the  public.  The  first  public  exhibition  of  the  completed  instru- 
ment was  made  in  1837  at  the  College  of  the  City  of  New  York, 
signals  being  sent  through  1700  feet  of  copper  wire.  It  was  with 
the  aid  of  a  $30,000  grant  from  Congress  that  the  first  commercial 
line  was  constructed  in  1844  between  Washington  and  Baltimore 


MAGNETIC  PKOPEKTIES  OF  COILS 


277 


Armature  Contact  Points 


Electro. 


armature  are  not  sufficiently  loud  to  be  read  easily  by  an  operator. 
Hence  at  each  station  there  is  introduced  a  local  circuit,  which  contains 
a  local  battery,  and  a  second  and  heavier  electromagnet,  which  is  called 
a  sounder.  The  electro- 
magnet on  the  main  line 
is  then  called  the  relay 
(see  Figs.  292,  293,  and 
294).  The  sounder  has 
a  very  heavy  armature 
(A,  Fig.  293),  which  is 
so  arranged  that  it  clicks 
both  when  it  is  drawn 


Spring 

Adjusting  Screw 


down  by  its  electromag- 


FIG.  292.   The  relay 


net   against   the    stop    S     • 
and  when  it  is  pushed  up  again  by  its  spring,  on  breaking  the  current^- 
against  the  stop  t.   The  interval  which  elapses  between  these  two  clicks 
indicates  to  the  operator  whether  a  dot  or  dash 
is  sent.    The  current  in  the  main  line  simply 
serves  to   close   and  open  the  circuit  in  the 
local  battery  which  operates  the  sounder  (see 
Fig.  294).    The   electromagnets   of  the  relay 
and   the    sounder    differ   in   that  the  former 
consists  of  many  thousand  turns  of  fine  wire, 
usually  having  a  resistance  of  about  150  ohms, 
while  the    latter   consists   of    a   few  hundred 
turns  of  coarse  wire  having  ordinarily  a  resistance  of  about  4  ohms. 
348.  Plan  of  a  telegraphic  system.    The  actual  arrangement  of  the 

various  parts  of  a  tele- 

Chicago 


FIG.  293.   The  sounder 


Sounder 


Sounder 


Neribrk 


graphic  system  is  shown 
in  Fig.  294.  When  an  op- 
erator at  Chicago  wishes 
to  send  a  message  to 
New  York,  he  first  opens 
the  switch  which  is  con- 
nected to  his  key,  and 
which  is  always  kept 
closed  except  when  he 
is  sending  a  message. 

He  then  begins  to  operate  his  key,  thus  controlling  the  clicks  of  both 
his  own  sounder  and  that  at  New  York.  When  the  Chicago  switch  is 
closed  and  the  one  at  New  York  open,  the  New  York  operator  is  able  to 


Earth  Earth, 

FIG.  294.   Telegraphic  system 


278        EFFECTS  OF  THE  ELECTRIC  CURRENT 

send  a  message  back  over  the  same  line.  In  practice  a  message  is  not 
usually  sent  as  far  as  from  Chicago  to  New  York  over  a  single  line, 
save  in  the  case  of  transoceanic  cables.  Instead  it  is  automatically 
transferred  at,  say,  Cleveland  to  a  second  line,  which  carries  it  on  to 
Buffalo,  where  it  is  again  transferred  to  a  third  line,  which  carries 
it  on  to  New  York.  The  transfer  is  made  in  precisely  the  same  way 
as  the  transfer  from  the  main  circuit  to  the  sounder  circuit.  If,  for 
example,  the  sounder  circuit  at  Cleveland  is  lengthened  so  as  to  extend 
to  Buffalo,  and  if  the  sounder  itself  is  replaced  by  a  relay  (called  in 
this  case  a  repeater),  and  the  local  battery  by  a  line  battery,  then  the 
sounder  circuit  has  been  transformed  into  a  repeater  circuit  and  all  the 
conditions  are  met  for  an  automatic  transfer  of  the  message  at  Cleveland. 

QUESTIONS  AND  PROBLEMS 

1.  Why  are  iron  wires  used  on  telegraph  lines  but  copper  wires  on 
trolley  systems? 

2.  The  plane  of  a  suspended  loop  of  wire  is  east  and  west.    A  cur- 
rent is  sent  through  it,  passing  from  east  to  west  on  the  upper  side. 
What  will  happen  to  the  loop  if  it  is  perfectly  free  to  turn  ? 

3.  When  a  strong  current  is  sent  through  a  suspended-coil  galva- 
nometer, what  position  will  the  coil  assume  ? 

4.  If  one  looks  down  on  the  ends  of  a  U-shaped  electromagnet,  does 
the  current  encircle  the  two  coils  in  the  same  or  in  opposite  directions  ? 
Does  it  run  clockwise  or  counterclockwise  about  the  JV  pole  ? 

5.  Draw  a  diagram,  showing  how  an  electric  bell  works. 

6.  Draw  a  diagram,  showing  how  the  relay  and  sounder  operate  in 
a  telegraphic  circuit. 

7.  Ordinary  No.  9  telegraph  wire  has  a  resistance  of  20  ohms  to 
the  mile.    What  current  will  100  Daniell  cells,  each  of  E.M.F.  of  1  volt, 
send  through  100  miles  of  such  wire,  if  the  relays  have  a  resistance  of 
150  ohms  each  and  the  cells  an  internal  resistance  of  4  ohms  each  ? 

8.  If  the  relays  of  the  preceding  problem  had  each  10,000  turns  of 
wire  in  their  coils,  how  many  ampere  turns  were  effective  in  magnetizing 
their  electromagnets  ? 

9.  If  on  the  above  telegraph  line  sounders  having  a  resistance  of 
3  ohms  each  and  500  turns  were  to  be  put  in  the  place  of  the  relays, 
how  many  ampere  turns  would  be  effective  in  magnetizing  their  cores  ? 
Why,  then,  must  a  relay  be  a  high-resistance  electromagnet  ? 

10.  If  the  earth's  magnetism  is  due  to  a  surface  charge  rotating  with 
the  earth,  must  this  charge  be  positive  or  negative  to  produce  the  sort 
of  magnetic  poles  which  the  earth  has?  (This  is  actually  the  present 
theory  of  the  earth's  magnetism.) 


HEATING  EFFECTS  279 

HEATING  EFFECTS  OF  THE  ELECTRIC  CURRENT 

349.  Heat  developed  in  a  wire  by  an  electric  current.   Let  the 

terminals  of  two  or  three  dry  cells  in  series  be  touched  to  a  piece  of  No.  40 
iron  or  German-silver  wire  and  the  length  of  wire  between  these  termi- 
nals shortened  to  1  inch  or  less.  The  wire  will  be  heated  to  incandes- 
cence and  probably  melted. 

The  experiment  shows  that  just  as  in  the  charging  of  a  stor- 
age battery  the  energy  of  the  electric  current  was  transformed 
into  the  energy  of  chemical  separation,  so  here  in  the  passage 
of  the  current  through  the  wire  the  energy  of  the  electric 
current  is  transformed  into  heat  energy. 

350.  Energy  relations  of  the  electric  current.    In  Chapter 
IX  we  found  that  energy  expended  on  a  water  turbine  is  equal 
to  the  quantity  of  water  passing  through  it  times  the  differ- 
ence in  level  through  which  the  water  falls.    In  just  the  same 
way  it   is  found   that  when   a  current  of  electricity  passes 
through  a  conductor,  the  energy  expended  is  equal  to  the 
quantity  of  electricity  passing  times  the  difference  in  potential 
between  the  ends  of  the  conductor.    If  the  quantity  of  elec- 
tricity is  expressed  in  coulombs  and  the  P.D.  in  volts,  the 
energy  is  given  in  joules,  and  we  have 

Volts  x  coulombs  =  joules.  (1) 

Since  the  number  of  coulombs  is  equal  to  the  number  of 
amperes  of  current  multiplied  by  the  number  of  seconds, 

Volts  x  amperes  x  seconds  =  joules.  (2) 

But  a  watt  is  denned  as  a  joule  per  second  (see  §  192).  Hence 
the  energy  expended  per  second  by  the  current,  that  is,  the 
power  of  the  current,  is  given  by 

Volts  x  amperes  =  watts.  (3) 

351.  Calories  of  heat  developed  in  a  wire.    The  electrical 
energy  expended  when  a  current  flows  between  points  of  given 
P.D.  may  be  spent  in  a  variety  of  ways.    For  example,  it  may 


280         EFFECTS  OF  THE  ELECTRIC  CURRENT 

be  spent  in  producing  chemical  separation,  as  in  the  charging 
of  a  storage  cell ;  it  may  be  spent  •  in  doing  mechanical  work, 
as  is  the  case  when  the  current  flows  through  an  electric  motor; 
or  it  may  be  spent  wholly  in  heating  the  wire,  as  was  the  case 
in  the  experiment  of  §  349.  It  will  always  be  expended  in 
this  last  way  when  no  chemical  or  mechanical  changes  are 
produced  by  it.  The  number  of  calories  of  heat  produced  per 
second  in  the  wire  of  the  last  experiment  is  found,  then,  by 
multiplying  the  number  of  joules  expended  by  the  current  per 
second  by  the  heat  equivalent  of  the  joule  in  calories,  that  is, 
.24  calorie,  since,  by  §  173  and  §  213,  1  calorie  is  4.2  joules. 
Therefore  when  all  of  the  electrical  energy  of  a  current  is 
transformed  into  heat  energy,  we  have 

Calories  per  second  =  volts  X  amperes  x  .24.          (4) 

The  total  number  of  calories  H  developed  in  t  seconds  will 
be  given  by  ff  =  pj)>  xCxtx  _24_  (5) 

Thus  a  current  of  10  amperes  flowing  in  a  wire  whose  ter- 
minals are  at  a  potential  difference  of  12  volts  will  develop 
in  5  minutes  10  x  12  x  300  x  .24  =  8640  calories. 

Since  by  Ohm's  law  P.D.  =  C  X  R,  we  have,  by  substitut- 
ing CR  for  P.D.  hi  (5), 

//=  C'2E  x  tx  .24;  (6) 

or  the  heat  generated  in  a  conductor  is  proportional  to  the  time, 
to  the  resistance,  and  to  the  square  of  the  current.  This  is  known 
as  Joule's  law,  having  been  first  announced  by  him  as  the 
result  of  experimental  researches. 

352.  Incandescent  lamps.  The  ordinary  incandescent  lamp 
consists  of  a  carbon  filament  heated  to  incandescence  by  an 
electric  current  (Fig.  295).  Since  the  carbon  would  burn  in- 
stantly in  air,  the  filament  is  placed  in  a  highly  exhausted  glass 
bulb.  Even  then  it  disintegrates  slowly.  The  normal  life  of  a 
16-candle-power  lamp  filament  is  from  1000  to  2000  working 


HEATING  EFFECTS  281 

hours.  The  filament  is  made  by  carbonizing  a  special  form  of 
cotton  thread.  The  ends  of  the  carbonized  thread  are  attached 
to  platinum  wires  which  are  sealed  into  the  glass  walls  of  the 
bulb,  and  which  make  contact  one  with  the  base  of  the  socket 
and  the  other  with  its  rim,  these  being  the 
electrodes  through  which  the  current  enters 
and  leaves  the  lamp. 

The  ordinary  16 -candle-power  lamp  is  most 
commonly  run  on  a  circuit  which  maintains  a 
potential  difference  of  either  110  or  220  volts 
between  the  terminals  of  the  lamp.  In  the  FIG.  295.  The  in- 
former case  the  lamp  carries  about  .5  ampere  candescent  lamP 
of  current,  and  in  the  latter  case  about  .25  ampere.  It  will  be 
seen  from  these  figures  that  the  rate  of  consumption  of  energy 
is  about  3.4  watts  per  candle  power. 

A  customer  usually  pays  for  his  light  by  the  "watt  hour," 
a  watt  hour  being  the  energy  furnished  in  one  hour  by  a 
current  whose  rate  of  expenditure  of  energy  is  one  watt. 
Thus  the  rate  at  which,  energy  is  consumed  by  a  16 -candle- 
power  lamp  is  110  x  .5  =  55  watts.  One  such  lamp  run- 
ning for  ten  hours  would  therefore  consume  550  watt  hours 
of  energy.  Large  quantities  of  electricity  are  sold  by  the 
kilowatt  hour. 

At  the  present  time  tungsten  and  tantalum  filaments  are 
being  used  very  largely  for  incandescent  lamps.  They  are 
nearly  three  times  as  efficient  as  the  carbon  lamps,  the  "  Mazda  " 
form  taking  but  1.25  watts  per  candle.  This  is  because  they 
can  be  operated  at  much  higher  temperatures  than  carbons. 
They  are,  however,  much  more  fragile  and  more  expensive. 

353.  The  arc  light.  When  two  carbon  rods  are  placed  end  to  end  in 
the  circuit  of  a  powerful  electric  generator,  the  carbon  about  the  point 
of  contact  is  heated  red-hot.  If,  then,  the  ends  of  the  carbon  rods  are 
separated  one-fourth  inch  or  so,  the  current  will  still  continue  to  flow, 
for  a  conducting  layer  of  incandescent  vapor  called  an  electric  arc  is 


28! 


EFFECTS  OF  THE  ELECTRIC  CURRENT 


produced  between  the  poles.    The  appearance  of  the  arc  is  shown  in 

Fig.  296.    At  the  +  pole  a  hollow,  or  crater,  is  formed  in  the  carbon, 

while  the  —  carbon  becomes  cone  shaped,  as  in  the 

figure.    The  carbons  are  consumed  at  the  rate  of 

about  an  inch  an  hour,  the  +  carbon  wasting  away 

about  twice  as  fast  as  the  —  one.    The  light  comes 

chiefty  from  the  +  crater,  where  the  temperature  is 

about  $800°  C.,  the  highest  attainable  by  man.    All 

known  substances  are  volatilized  in  the  electric  arc. 

The  ordinary  arc  requires  a  current  of  10  amperes 
and  a  P.D.  between  its  terminals  of  about  50  volts. 
Such  a  lamp  produces  about  500*  candle  power, 
and  therefore  consumes  energy  at  the  rate  of  about 
1  watt  per  candle  power.  This  makes  an  arc  light 
about  3.5  times  as  efficient  as  an  incandescent  light. 
The  recently  invented  flaming  arc,  produced  between 
carbons  which  have  a  composite  core  consisting  of 
carbon,  lime,  magnesia,  silica,  or  other  light-giving 
minerals,  sometimes  reaches  an  efficiency  as  high 
as  .27  watt  per  candle  power. 

354.  The  arc-light  automatic  feed.  Since  the  two  carbons  of  the  arc 
gradually  waste  away,  they  would  soon  become  so  far  separated  that 
the  arc  could  no  longer  be  maintained  were 
it  not  for  an  automatic  feeding  device  which 
keeps  the  distance  between  the  carbon  tips 
very  nearly  constant.  Fig.  297  shows  the  es- 
sential features  of  one  form  of  this  device. 
When  no  current  is  flowing  through  the 
lamp,  gravity  holds  the  carbon  tips  at  e  to- 
gether ;  but  as  soon  as  the  current  is  thrown 
on,  it  energizes  the  low-resistance  electro- 
magnet M,  which  is  in  series  with  the  car- 
bons. This  draws  down  the  iron  plunger  c, 
which  acts  upon  the  lever  L  and  "  strikes  the 
arc  "  at  e.  But  the  introduction  of  the  resist- 


FIG.  296.  The  arc 
light 


FIG.  297.    Feeding  device 
for  arc  lamp 


ance  of  the  arc  into  the  circuit  sABMt  raises  the  P.I),  between  s  and  t  and 
thus  causes  an  appreciable  current  to  flow  through  the  high-resistance 
magnet  JV,  which  is  shunted  across  this  circuit.  This  tends  to  raise  th<- 
plunger  c  and  thus  to  shorten  the  arc.  There  is  thus  one  particular  leng^ 

*This  is  the  so-called  "mean  spherical  "  candle  power.   The  candle  powei 
the  direction  of  maximum  illumination  is  from  1000  to  1200. 


HEATING  EFFECTS  283 

,w»Jk»  tr  r>  \ 

of  arc  for  whiclf  equilibrium  exists  between  the  "effects  of  the  series  magnet 

M  and  the  shunt  magnet  N.  This  length  the  lamp  automatically  main- 
tains. The  magnet  0  is  the  so-called  "  cut-out "  inserted  so  that,  if  the  lamp 
gets  out  of  order  and  the  arc  burns  out,  a  current  at 
once  flows  through  N  and  0  of  sufficient  strength  to 
close  the  contact  points  at  r  and  thus  permit  the  main 
current  to  flow  on  to  the  next  lamp  over  the  path  PrR  Q. 
355.  The  Cooper-Hewitt  mercury  lamp.  The  Cooper- 
Hewitt  mercury  lamp  (Fig.  298)  is  the  most  efficient 
of  all  electric  lights,  unless  it  be  the  flaming  arc.  It 
differs  from  the  arc  lamp  in  that  the  incandescent  body 
is  a  long  column  of  mercury  vapor  instead  of  an  incan- 
descent solid.  The  lamp  consists  of  an  exhausted  tube 
three  or  four  feet  long,  the  positive  electrode  at  the  top 
consisting  of  a  plate  of  iron,  while  the  negative  elec- 
trode at  the  bottom  is  a  small  quantity  of  mercury. 
Under  a  sufficient  difference  of  potential  between  these 
terminals  a  long  mercury-vapor  arc  is  formed  which 
stretches  from  terminal  to  terminal  in  the  tube.  This 
arc  emits  a  very  brilliant  light,  but  it  is  almost  entirely 
wanting  in  red  rays.  The  efficiency  of  the  lamp  is  very 
high,  since  it  requires  but  .3  watt  per  candle  power.  It 
is  rapidly  finding  important  commercial  uses,  especially  in  photogra- 
phy. The  chief  objection  to  it  arises  from  the  fact  that,  on  account 
of  the  absence  of  red  rays,  the  light  gives  objects  an  unnatural  color. 

QUESTIONS  AND  PROBLEMS 

••iJ0i 

1.  What  horse  power  is  required  to  run  an  incandescent  lamp  carry- 
ing .5  ampere  at  110  volts  ?    How  much  heat  is  developed  per  second  ? 

2.  Fig.  299   shows  the  connections  for  a  lamp  L  which  can   be 
turned  on  or  off  at  two  different 

points   a  or  &.   Explain  how  it 
works. 

3.  A  220-volt  lamp  has  a  re- 
sistance, when  hot,  of  about  750 
ohms.    How  many  calories  will 
be  developed  in  it  in  10  min.  ? 

4.  If   a  storage  cell  has  an  E.M.F.  of  2  volts,  and  furnishes  a  cur- 
rent of  5  amperes,  what  is  its  rate  of  expenditure  of  energy  in  watts  ? 

5.  How  many  cells,  working  as  in  Problem  4,  would  be  equivalent  to 
1H.P.?   (See  §191,  p.  147.) 


FIG.  289.  The 
Cooper-Hewitt 
mercury  lamp 


FIG.  299 


<7 


CHAPTER  XV 

INDUCED  CURRENTS 
THE  PRINCIPLE  OF  THE  DYNAMO  AND  MOTOR 

356.  Current  induced  by  a  magnet.  Let  400  or  500  turns  of 
No.  22  copper  wire  be  wound  into  a  coil  C  (Fig.  300)  about  two  and  a 
half  inches  in  diameter.  Let  this  coil  be  connected  into  circuit  with  a 
lecture-table  galvanometer  (Fig.  254),  or  even  a  simple  detector  made  by 
suspending  in  a  box,  with 
No.  40  copper  wire,  a  coil 
of  200  turns  of  No.  30 
copper  wire  (see  Fig.  300). 
Let  the  coil  C  be  thrust 
suddenly  over  the  N  pole 
of  a  strong  horseshoe  mag- 
net. The  deflection  of  the  ^IG  300  ^Induction  of  electric  currents  by 
pointer  p  of  the  galva-  magnets 

iiometer    will    indicate    a 

momentary  current  flowing  through  the  coil.  Let  the  coil  be  held  sta- 
tionary over  the  magnet.  The  pointer  will  be  found  to  come  to  rest  in  its 
natural  position.  Now  let  the  coil  be  removed  suddenly  from  the  pole. 
The  pointer  will  move  in  a  direction  opposite  to  that  of  its  first  deflec- 
tion, showing  that  a  reverse  current  is  now  being  generated  in  the  coil. 

We  learn,  therefore,  that  a  current  of  electricity  may  be 
induced  in  a  conductor  by  causing  the  latter  to  move  through  a 
magnetic  field,  while  a  magnet  has  no  such  influence  upon  a 
conductor  which  is  at  rest  with  respect  to  the  field.  This  dis- 
covery, one  of  the  most  important  in  the  history  of  science, 
was  announced  by  the  great  Faraday  in  1831.  From  it  have 
sprung  directly  most  of  the  modern  industrial  developments 
of  electricity. 

284 


MICHAEL  FARADAY  (1791-1867) 

Famous  English  physicist  and  chemist ;  one  of  the  most  gifted  of  experimenters ; 
son  of  a  poor  blacksmith ;  apprenticed  at  the  age  of  thirteen  to  a  London  book- 
hinder,  with  whom  he  worked  nine  years ;  applied  for  a  position  in  Sir  Humphry 
Davy's  laboratory  at  the  Royal  Institution  in  1813 ;  became  director  of  this  labo- 
ratory in  1825 ;  discovered  electromagnetic  induction  in  1831 ;  made  the  first 
dynamo ;  discovered  in  1833  the  laws  of  electrolysis,  now  known  as  Faraday's 
laws ;  the  farad,  the  practical  unit  of  electrical  capacity,  is  named  in  his  honor 


PRINCIPLE  OF  THE  DYNAMO  AND  MOTOR     285 


357.  Direction  of  induced  current.    Lenz's  law.    In  order  to 

find  the  direction  of  the  induced  current,  let  a  very  small  P.D.  from  a 
galvanic  cell  be  applied  to  the  terminals  A  and  B  (Fig.  300),  and  note 
the  direction  in  which  the  pointer  moves  when  the  current  enters,  say 
at  A.  This  will  at  once  show  in  what  direction  the  current  was  flow- 
ing in  the  coil  C  when  it  was  being  thrust  over  the  Arpole.  By  a  simple 
application  to  C  of  the  right-hand  rule  (§  342)  we  can  then  tell  which 
was  the  N  and  which  the  S  face  of  the  coil  when  the  induced  current 
was  flowing  through  it.  In  this  way  it  will  be  found  that  if  the  coil  was 
being  moved  past  the  N  pole  of  the  magnet,  the  current  induced  in  it 
was  in  such  a  direction  as  to  make  the  lower  face  of  the  coil  an  N  pole 
during  the  downward  motion  and  an  S  pole  during  the  upward  motion. 
In  the  first  case  the  repulsion  of  the  N  pole  of  the  magnet  and  the  N 
pole  of  the  coil  tended  to  oppose  the  motion  of  the  coil  while  it  was 
moving  from  a  to  I,  and  the  attraction  of  the  N  pole  of  the  magnet  and 
the  S  pole  of  the  coil  tended  to  oppose  the  motion  while  it  was  moving 
from  b  to  c.  In  the  second  case  the  repulsion  of  the  two  JV  poles  tended 
to  oppose  the  motion  between  b  and  c,  and  the  attraction  between  the 
N  pole  of  the  magnet  and  the  S  pole  of  the  coil  tended  to  oppose  the 
upward  motion  from  b  to  a.  In  every  case,  therefore,  the  motion  is  made 
against  (in  opposing  force. 

From  these  experiments,  and  others  like  them,  we  arrive  at 
the  following  law :  Whenever  a  current  is  induced  by  the  rela- 
tive motion  of  a  magnetic  field  and  a  conductor,  the  direction  of 
the  induced  current  is  always  such  as  to  set>  up  a  magnetic  field 
which  opposes  the  motion.  This  is  Lenz's  law. 
This  law  might  have  been  predicted  at  once 
from  the  principle  of  the  conservation  of 
energy ;  for  this  principle  tells  us  that  since 
an  electric  current  possesses  energy,  such 
a  current  can  appear  only  through  the 
expenditure  of  mechanical  work  or  of  some 
other  form  of  energy. 

358.  Condition  necessary  for  an  induced 
E.M.F.    Let  the  coil  be  held  in  the   position 

shown  in  Fig.  301,  and  moved  back  and  forth  parallel  Ijp  the  magnetic 
field ;  that  is,  parallel  to  the  line  NS.  No  current  will  be  induced. 


FIG.  301.    Currents 

induced   only  when 

conductor  cuts  lines 

of  force 


286 


INDUCED  CURRENTS 


FIG.  302.    E.M.F. 

induced  when  a 
straight  conductor 
cuts  magnetic  lines 


By  experiments  of  this  sort  it  is  found  that  an  E.M.F.  is 
induced  in  a  coil  only  when  the  motion  takes  place  in  such  a  way 
as  to  change  the  total  number  of  magnetic  lines  t 

of  force  which  are  inclosed  by  the  coil.  Or,  to 
state  this  rule  in  more  general  form,  an 
E.M.F.  is  induced  in  any  element  of  a  con- 
ductor when,  and  only  ivhen,  that  element  is 
moving  in  such  a  way  as  to  cut  magnetic  lines 
of  force.* 

It  will  be  noticed  that  the  first  statement 
of  the  rule  is  included  in  the  second,  for 
whenever  the  number  of  lines  of  force  which 
pass  through  a  coil   changes,  some  lines  of  force  must  cut 
across  the  coil  from  the  inside  to  the  outside,  or  vice  versa. 

359.  The  principle  of  the  electric  motor. 

Let  a  vertical  wire  ab  be  rigidly  attached  to  a 
horizontal  wire  yli,  and  let  the  latter  be  supported 
by  a  ring  or  other  metallic  support,  in  the  manner 
shown  in  Fig.  303,  so  that  ab  is  free  to  oscillate 
about  (jh  as  an  axis.  Let  the  lower  end  of  ab  dip 
into  a  trough  of  mercury.  When  a  magnet  is  held 
in  the  position  shown  and  a  current  from  a  dry 
cell  is  sent  down  through  the  wire,  the  wire  will 
instantly  move  in  the  direction  indicated  by  the 
arrow  /,  namely,  at  right  angles  to  the  direction 
of  the  lines  of  magnetic  force.  Let  the  direction 
of  the  current  in  the  wire  be  reversed.  The  direc- 
tion of  the  force  acting  on  the  wire  will  be  found 
to  be  reversed  also. 

FIG.  303.  The  prin- 

We  learn,  therefore,  that  a  wire  carrying  ciple  of  the  electric 
a  current  in  a  magnetic  field  tends  to  move  in  motor 

*  If  a  strong  electromagnet  is  available,  these  experiments  are  more  instructive 
if  performed,  not  with  a  coil,  as  in  Fig.  301,  but  with  a  straight  rod  (Fig.  302) 
to  the  ends  of  which  are  attached  wires  leading  to  a  galvanometer.  When- 
ever the  rod  moves  parallel  to  the  lines  of  magnetic  force  there  will  be  no 
deflection,  but  whenever  it  moves  across  the  lines  the  galvanometer  needle  will 
move  at  once. 


PRINCIPLE  OF  THE  DYNAMO  AND  MOTOR     287 

a  direction  at  right  angles  both  to  the  direction  of  the  field  and 
the  direction  of  the  current.  This  fact  underlies  the  operation 
of  all  electric  motors. 

360.  The  motor  and  dynamo  rules.  A  convenient  rule  for 
determining  whether  the  wire  ab  (Fig,  303)  will  move  forward 
or  back  in  a  given  case  may  be  obtained  as  follows :  If  the 
field  of  a  magnet  alone  is  represented  by  Fig.  304,  and  that 
due  to  the  current  *  alone  by  Fig.  305,  then  the  resultant  field 


FIG.  304.   Field  of 
magnet  alone 


FIG.  305.    Field  of 
current  alone 


FIG.  306.  Field  of  magnet 
and  current 


when  the  current-bearing  wire  is  placed  between  the  poles  of 
the  magnet  is  that  shown  in  Fig.  306  ;  for  the  strength  of  the 
field  above  the  wire  is  now  the  sum  of  the  two  separate  fields, 
while  the  strength  below  it  is  their  difference.  Now  Faraday 
thought  of  the  lines  of  force  as  acting  lilie  stretched  rubber 
bands.  This  would  mean  that  the  wire  in  Fig.  306  would  be 
pushed  down.  Whether  the  lines  of  force  are  so  conceived  or 
not,  the  motor  rule  may  be  stated  thus : 

A  current  in  a  magnetic  field  tends  to  move  away  from  the  side 
on  which  its  lines  are  added  to  those  of  the  field. 

The  dynamo  rule  follows  at  once  from  the  motor  rule  and 
Lenz's  law.  Thus  when  a  wire  is  moved  through  a  magnetic 
field  the  current  induced  in  it  must  be  in  such  a  direction  as 

*  The  cross  in  the  conductor  of  Fig.  305,  representing  the  tail  of  a  retreating 
arrow,  is  to  indicate  that  the  current  flows  away  from  the  reader.  A  dot,  represent- 
ing the  head  of  an  advancing  arrow,  indicates  a  current  flowing  toward  the  reader. 


288  INDUCED  CURRENTS 

to  oppose  the  motion ;  therefore  the  induced  current  will  be  in 
such  a  direction  as  to  increase  the  number  of  lines  on  the  side 
toward  which  it  is  moving.* 

361.  Strength  of  the  induced  E.M.F.    The  strength  of  an 
induced  E.M.F.  is  found  to  depend  simply  upon  the  number  of 
lines  of  force  cut  per  second  by  the  conductor,  or,  in  the  case 
of  a  coil,  upon  the  rate  of  change  in  the  number  of  lines  of 
force  which  pass  through  the  coil.    The  strength  of  the  current 
which  flows  is  then  given  by  Ohm's  law  ;  that  is,  it  is  equal  to 
the  induced  E.M.F.  divided  by  the  resistance  of  the  circuit. 
The  number  of  lines  of  force  which  the  conductor  cuts  per 
second  may  always  be  determined  if  we  know  the  velocity  of 
the  conductor  and  the  strength  of  the  magnetic  field'  through 
which  it  moves.   For  it  will  be  remembered  that,  according  to 
the  convention  of  §  275,  a  field  of  unit  strength  is  said  to  con- 
tain one  line  of  force  per  square  centimeter,  a  field  of  1000 
units  strength  1000  lines  per  square  centimeter,  etc.    In  a 
conductor  which  is  cutting  lines  at  the  rate  of  100,000,000 
per  second  there  is  an  induced  E.M.F.  of  1  volt.t    The  reason 
that  we  used  a  coil  of  500  turns  instead  of  a  single  turn  in  the 
experiment  of  §  356  was  that  by  thus  making  the  conductor 
in  which  the  current  was  to  be  induced  cut  the  lines  of  force 
of  the  magnet  500  times  instead  of  once,  we  obtained  500 
times  as  strong  an  induced  E.M.F.,  and  therefore  500  times 
as  strong  a  current  for  a  given  resistance  in  the  circuit. 

362.  Currents  induced  in  rotating  coils.   Let  a  400-  or  500-tum 

coil  of  No.  28  copper  wire  be  made  small  enough  to  rotate  between  the 
poles  of  a  horseshoe  magnet,  and  let  it  be  connected  into  the  circuit  of 
a  galvanometer,  precisely  as  in  §  356.  Starting  with  the  coil  in  the  posi- 
tion of  Fig.  307,  (1),  let  it  be  rotated  suddenly  clockwise  (looking  down 

*This  may  be  thrown  into  a  convenient  rule  of  thumb  as  follows:  "  Grasp  the 
conductor  with  the  right  hand,  the  fingers  extending  in  the  direction  of  the  lines 
of  force  to  be  cut;  the  thumb  will  indicate  the  direction  of  the  induced  current." 

t  This  may  be  considered  as  the  scientific  definition  of  the  volt,  convenience  alone 
having  dictated  the  legal  definition  given  in  §  331. 


PRINCIPLE  OF  THE  DYXAMO  AND  MOTOR     289 


FIG.  307.     Direction  of  cur- 
rents induced  in  a  coil  rotat- 
ing in  a  magnetic  field 


from  above)  through  180°.  A  strong  deflection  of  the  galvanometer  will 
be  observed.  Let  it  be  rotated  through  the  next  180°  back  to  the  starting 
point.  An  opposite  deflection  will  be  observed. 

The  arrangement  is  a  dynamo  in  miniature.  During  the  first 
half  of  the  revolution  [see  Fig.  307,  (2)]  the  wires  on  the  right 
side  of  the  loop  were  cutting  the  lines  of  force  in  one  direc- 
tion, while  the  wires  on  the  left  side  were  cutting  them  in  the 
opposite  direction.  A  current  was 
being  generated  down  on  the  right 
side  of  the  coil  and  up  on  the  left 
side  (see  dynamo  rule).  It  will  be 
seen  that  both  currents  flow  around 
the  coil  in  the  same  direction.  The 
induced  current  is  strongest  when 
the  coil  is  in  the  position  shown  in 
Fig.  307,  (2),  because  there  the  lines 
of  force  are  being  cut  most  rapidly. 
Just  as  the  coil  is  moving  into  or 

out  of  the  position  shown  in  Fig.  307,  (1),  it  is  moving  parallel 
to  the  lines  of  force,  and  hence  no  current  is  induced,  since  no 
lines  of  force  are  being  cut.  As  the  coil  moves  through  the 
last  180°  of  its  revolution  both  sides  are  cutting  the  same  lines 
of  force  as  before,  but  they  are  cutting  them  in  an  opposite 
direction ;  hence  the  current  generated  during  this  last  half  is 
opposite  in  direction  to  that  of  the  first  half.* 

QUESTIONS  AND  PROBLEMS 

1.  Under  what  conditions  may  an  electric  current  be  produced  by  a 
magnet  ? 

2.  A  current  is  flowing  from  top  to  bottom  in  a  vertical  wire.    In 
what  direction  will  the  wire  tend  to  move  on  account  of  the  earth's 
magnetic  field? 

3.  State  Lenz's  law,  and  show  how  it  follows  from  the  principle  of  the 
conservation  of  energy. 

*  A  laboratory  experiment  on  the  principles  of  induction  should  be  performed 
at  about  this  point.   See,  for  example,  Experiment  .'36  of  the  authors'  manual. 


290 


INDUCED  CURRENTS 


4.  If  the  coil  of  a  sensitive  galvanometer  is  set  to  swinging  while 
the  circuit  through  the  coil  is  open,  it  will  continue  to  swing  for  a  long 
time ;  but  if  the  coil  is  short-circuited,  it  will  come  to  rest  after  a  very 
few  oscillations.   Why  ?   (The  experiment  may  easily  be  tried.  Remem- 
ber that  currents  are  induced  in  the  moving  coil.    Apply  Lenz's  law.) 

5.  A  coil  is  thrust  over  the  S  pole  of  a  magnet.  Is  the  direction  of 
the  induced  current  clockwise  or  counterclockwise  as  you  look  down 
upon  the  pole  ? 

6.  A  ship  having  an  iron  mast  is  sailing  east.    In  what  direction  is 
the  E.M.F.  induced  in  the  mast  by  the  earth's  magnetic  field  ?  If  a  wire 
is  brought  from  the  top  of  the  mast  to  its  bottom,  no  current  will  flow 
through  the  circuit.    Why  ? 

7.  When  a  wire  is  cutting  lines  of  force  at  the  rate  of  100,000,000 
per  second,  there  is  induced  in  it  an  E.M.F.  of  one  volt.    A  certain 
dynamo  armature  has  50  coils  of  5  loops  each  and  makes  600  revolutions 
per  minute.  Each  wire  cuts  2,000,000  lines  of  force  twice  in  a  revolution. 
What  is  the  E.M.F.  developed  ? 

8.  If  a  coil  of  wire  is  rotated  about  a  vertical  axis  in  the  earth's 
field,  an  alternating  current  is  set  up  in  it.    In  what  position  is  the  coil 
when  the  current  changes  direction  ? 


DYNAMOS 

363.  A  simple  alternating-current  dynamo.  The  simplest 
form  of  commercial  dynamo  consists  of  a  coil  of  wire  so  arranged 
as  to  rotate  continuously  between  the  poles  of  a  powerful 
electromagnet  (Fig.  308). 

In  order  to  make  the  mag- 
netic field  in  which  the  con- 
ductor is  moved  as  strong  as 
possible,  the  coil  is  wound 
upon  an  iron  core  O.  This 
greatly  increases  the  total 
number  of  lines  of  magnetic 
force  which  pass  between  N 
and  $,  for  the  core  offers  an 
iron  path,  as  shown  in  Fig. 
309,  instead  of  an  air  path.  FlG.  308.  King-wound  armature 


DYNAMOS 


291 


FIG.  309.   End  view  of  ring 
armature 


The  rotating  part,  consisting  of  the  coil  with  its  core,  is 
called  the  armature.  If  the  coil  is  wound  in  the  manner  shown 
in  Figs.  308  and  309,  the  armature 
is  said  to  be  of  the  ring  type;  if 
in  the  manner  shown  in  Figs.  310 
and  311,  it  is  said  to  be  of  the 
drum  type.  The  latter  form  of 
winding  is  used  almost  exclusively 
in  modern  machines. 

One  end  of  the  coil  is  attached 
to  the  insulated  metal  ring  R,  which 
is  attached  rigidly  to  the  shaft  of 
the  armature  and  therefore  rotates  with  it,  while  the  other  end 
of  the  coil  is  attached  to  a  second  ring  R'.  The  brushes  b  and  V, 
which  constitute  the  terminals  of 
the  external  circuit,  are  always 
in  contact  with  these  rings. 

As  the  coil  rotates  an  in- 
duced alternating  current  passes 
through  the  circuit.  This  cur- 
rent reverses  direction  as  often 
as  the  coil  passes  through  the 
position  shown  in  Figs.  309  and 
311,  that  is,  the  position  in  which  the  conductors  are  moving 
parallel  to  the  lines  of  force ;  for  at  this  instant  the  conductors 
which  were  moving  up  begin  to 
move  down,  and  those  which  were 
moving  down  begin  to  move  up. 
The  current  reaches  its  maximum 
value  when  the  coils  are  moving 
through  a  position  90°  farther  on, 
for  then  the  lines  of  force  are 
being  cut  most  rapidly  by  the  con- 
ductors on  both  sides  of  the  coil. 

t 


FIG.  310.   Drum-wound  armature 


FIG.  311. 


End  view  of  drum 
armature 


292 


INDUCED  CURRENTS 


FIG.  312.     Diagram  of  alternating- 
current  dynamo 


364.  The  multipolar  alternator.    For  most  commercial  purposes  it  is 
found  desirable  to  have  120  or  more  alternations  of  current  per  second. 
This  could  not  be  attained  easily  with  two-pole  machines  like  those 
sketched  in  Figs.  303  to  311.  Hence 

commercial  alternators  are  usually 
built  with  a  large  number  of  poles 
alternately  N  and  S,  arranged 
around  the  circumference  of  a 
circle  in  the  manner  shown  in 
Fig.  312.  The  dotted  lines  repre- 
sent the  direction  of  the  lines  of 
force  through  the  iron.  It  will  be 
seen  that  the  coils  which  are  pass- 
ing beneath  JV  poles  have  induced 
currents  set  up  in  them,  the  direc- 
tion of  which  is  opposite  to  that 
of  the  currents  which  are  induced 
in  the  conductors  which  are  passing  beneath  the  S  poles.  Since,  how- 
ever, the  direction  of  winding  of  the  armature  coils  changes  between 
each  two  poles,  all  the  inductive  effects  of  all  the  poles  are  added  in  the 
coil  and  constitute  at  any  instant  one  single  current  flowing  around  the 
complete  circuit  in.  the  manner  indicated  by  the  arrows  in  the  diagram. 
This  current  reverses  direc- 
tion at  the  instant  at  which 
all  the  coils  pass  the  midway 
points  between  the  N  and  S 
poles.  The  number  of  altern,a- 
tions  per  second  is  equal  to 
the  number  of  poles  multi- 
plied by  the  number  of  revo- 
lutions per  second.  The  field 
magnets  N  and  S  of  such  a 
dynamo  are  usually  excited 
by  a  .direct  current  from  some 
other  source.  Fig.  313  rep- 
resents a  modern  commercial 
alternator.  FIG.  313.  Alternating-current  dynamo 

365.  The  principle  of  the  commutator.    By  the  use  of  a  so- 
called  commutator  it  is  possible  to  transform  a  current  which 


DYNAMOS 


293 


is  alternating  in  the  coils  of  the  armature  to  one  which  always 
flows  in  the  same  direction  through  the  external  portion  of 
the  circuit.  The  simplest  possible  form  of  such  a  commutator 
is  shown  in  Fig.  314.  It  consists  of  a  single  metallic  ring 
which  is  split  into  two  equal  in- 
sulated semicircular  segments 
a  and  c.  One  end  of  the  rotat- 
ing coil  is  soldered  to  one  of 
these  semicircles,  and  the  other 
end  to  the  other  semicircle. 
Brushes  b  and  b'  are  set  in 

such  positions  that  they  lose       FIG  314    The  simple  commutator 
contact   with    one    semicircle 

and  make  contact  with  the  other  at  the  instant  at  which  the 
current  changes  direction  in  the  armature.  The  current  there- 
fore always  passes  out  to  the  external  circuit  through  the 
same  brush.  While  a  current  from  such  a  coil  and  commu- 
tator as  that  shown  in  the  figure  would  always  flow  in  the 
same  direction  through  the  ex- 
ternal circuit,  it  would  be  of  a 
pulsating  rather  than  a  steady 
character,  for  it  would  rise  to 
a  maximum  and  fall  again  to 
zero  twice  during  each  complete 
revolution  of  the  armature.  This 
effect  is  avoided  in  the  commer- 
cial direct-current  dynamo  by 


FIG.  315.   Two-pole  direct-current 
dynamo  with  ring  armature 


building  a  commutator  of  a  large  number  of  segments  instead 
of  two,  and  connecting  each  to  a  portion  of  the  armature  coil 
in  the  manner  shown  in  Fig.  315. 

366.  The  ring-armature  direct-current  dynamo.  Fig.  315  is  a  diagram 
illustrating  the  construction  of  a  commercial  two-pole  direct-current 
dynamo  of  the  ring-armature  type.  The  figure  represents  an  end  view 
of  a  core  like  that  shown  in  Fig.  308.  The  coil  is  wound  continuously 


294 


INDUCED  CUftKENTS 


around  the  core,  each  segment  being  connected  to  a  corresponding  seg- 
ment of  the  commutator,  in  the  manner  shown  in  the  figure.  At  a  given 
instant  currents  are  being  induced  in  the  same  direction  in  all  the  con- 
ductors on  the  outside  of  the  core  on  the  left  half  of  the  armature.  The 
cross  on  these  conductors,  representing  the  tail  of  a  retreating  arrow,  is 
to  indicate  that  these  currents  flow  away  from  the  reader.  No  E.M.F.'s 
are  induced  in  the  conductors  on  the  inner  side  of  the  ring,  since  these 
conductors  cut  no  lines  of  force  (see  Fig.  309)  ;  nor  are  currents  induced 
in  the  conductors  at  the  top  and  bottom  of  the  ring  where  the  motion 
is  parallel  to  the  magnetic  lines.  The  addition  of  all  these  similarly 
directed  currents  in  the  various  convolutions  of  the  continuous  coil  on 
the  left  side  of  the  ring  constitutes  one  single  current  flowing  upward 
through  this  coil  toward  the  brush  b  (see  arrows).  On  the  right  half  of 
the  ring,  on  the  other  hand,  the  in- 
duced currents  are  all  in  the  opposite 
direction,  that  is,  toward  the  reader, 
since  the  conductors  are  here  all 
moving  up  instead  of  down.  The 
dot  in  the  middle  of  these  conductors 
represents  the  head  of  an  approach- 
ing arrow.  The  summation  of  these 
currents  constitutes  one  single  cur- 
rent also  flowing  upward  in  the  right 
half  of  the  coil  toward  the  brush  b. 
These  two  currents  from  the  two 
halves  of  the  ring  pass  out  at  b 
through  the  external  circuit  and  back  at  ?/.  This  condition  always 
exists,  no  matter  how  fast  the  rotation ;  for  it  will  be  seen  that  as  each 
loop  rotates  into  the  position  where  the  direction  of  its  current  reverses, 
it  passes  a  brush  and  therefore  at  once  becomes  a  part  of  the  circuit  on 
the  other  half  of  the  ring,  where  the  currents  are  all  flowing  in  the 
opposite  direction. 

If  the  machine  is  of  the  four-pole  type,  like  that  shown  in  Fig.  316, 
the  currents  flow  toward  two  neutral  points,  or  points  of  no  induction, 
instead  of  toward  one,  as  in  two-pole  machines,  and  they  flow  away  from 
two  other  neutral  points  (see  p,  p,  p',  p',  Fig.  316).  Hence  there  are 
four  brushes,  two  positive  and  two  negative,  as  in  the  figure.  Since  the 
two  positive  and  the  two  negative  brushes  are  connected  as  shown,  both 
sets  of  currents  flow  off  to  the  external  circuit  on  a  single  wire.  The 
figure  with  its  arrows  will  explain  completely  the  generation  of  currents 
by  a  four-pole  machine. 


FIG.  316.    Four-pole  direct-current 
dynamo,  ring-armature  type 


DYNAMOS 


295 


FIG.  317.   The  direct-current 
dynamo,  drum  winding 


367.  The    drum-armature  direct-current    dynamo.    The    drum-wound 
armature,  shown  in  section  in  Fig.  317,  has  an  advantage  over  the 
ring  armature  in  that,  while  the  con- 
ductors   on   the   inside    of   the    latter 

never  cut  lines  of  force  and  are  there- 
fore always  idle,  in  the  former  all  of  the 
conductors  are  cutting  lines  of  force 
except  when  they  are  passing  the  neu- 
tral points.  In  theory,  however,  the 
operation  of  the  drum  armature  is  pre- 
cisely the  same  as  that  of  the  ring 
armature.  All  the  conductors  on  the 
left  side  of  the  line  connecting  the 
brushes  (see  Fig.  317)  carry  induced 
currents  which  flow  in  one  direction,  while  all  the  conductors  on  the 

right  side  of  this  line  have 
opposite  currents  induced  in 
them.  It  will  be  seen,  how- 
ever, in  tracing  out  the  con- 
nections 1,  lp  2,  21?  3, 3P  etc., 
of  Fig.  317  (the  dotted  lines 
representing  connections  at 
the  back  of  the  drum),  that 
the  coil  is  so  wound  about  the 
drum  that  the  currents  in 
both  halves  are  always  flow- 
ing toVard  one  brush  &,  from 
which  they  are  led  to  the  ex- 
ternal circuit.  Fig.  318  shows 
a  typical  modern  four-pole 
FIG.  318.  Holtzer-Cabot  four-pole  direct-  generator,  and  Fig.  319  the 
current  generator  corresponding  drum-wound 

armature.    Fig.  329  (p.  305) 

illustrates*  nicely  the  method  of  winding  such  an  armature,  each  coil 
beginning  on   one   segment  of  the 
commutator    and    ending    on    the 
adjacent  segment. 

368.  Series-,  shunt-,  and  compound- 
wound  dynamos.     In   direct-current 
machines    the  field    magnet  NS  is 

excited  by  the  current  which   the        FIG.  319.   Holtzer-Cabot  armature 


296 


INDUCED  CURRENTS 


dynamo  itself  produces.  In  the  so-called  skunt-ivound  machines  a  small 
portion  of  the'current  is  led  off  from  the  brushes  through  a  great  many 
turns  of  fine  wire  which  encircle  the  core  of  the  magnet,  while  the  rest 
of  the  current  flows  through  the  external  circuit  (see  Fig.  320).  In  the  so- 
called  series-wound  dynamo  (Fig.  321)  the  whole  of  the  current  is  carried 
through  a  few  turns  of  coarse  wire  which  encircle  the  field  magnets. 
These  turns  are  then  in  series  with  the  external  circuit.  In  the  compound- 
wound  machine  (Fig.  322)  there  is  both  a  series  and  a  shunt  coil.  By 
this  arrangement  it  is  possible  to  maintain  a  constant  potential  differ- 
ence between  the  brushes,  no  matter  how  much  the  resistance  of  the 
external  circuit  may  be  varied.  Hence,  for  purposes  in  which  a  varying 


Main  Circuit 


Main  Circuit 


Main  Circuit 

X— X-X-X-N 


FIG.  320.    The   shunt- 
wound  dynamo 


FIG.  321.    The  series- 
wound  dynamo 


FIG.  322.  The  compound- 
wound  dynamo 


current  is  demanded,  as  in  incandescent  lighting,  the  operation  of  street 
cars,  etc.,  compound-wound  dynamos  are  almost  exclusively  used. 

Tn  all  these  types  of  self-exciting  machines  there  is  enough  residual 
magnetism  left  in  the  iron  cores  after  stopping  to  start  feeble  induced 
currents  when  started  up  again.  These  currents  immediately  increase 
the  strength  of  the  magnetic  field,  and  so  the  machine  quickly  builds 
up  its  current  until  the  limit  of  magnetization  is  reached. 

For  incandescent  electric  lighting  it  is  customary  to  use  a  dynamo 
of  the  compound  type  which  gives  a  P.D.  between  its  "  mains  "  of  either 
110  or  220  volts.  The  lamps  are  always  arranged  in  parallel  between 
these  mains,  as  is  illustrated  in  Fig.  322.  In  arc  lighting  a  series-wound 
dynamo  is  usually  used,  and  the  lamps  are  almost  invariably  arranged 
in  series,  as  in  Fig.  321.  About  50  lamps  are  commonly  fed  by  one 
machine.  This  requires  a  dynamo  capable  of  producing  a  voltage  of 
2500  volts,  since  each  lamp  requires  a  pressure  of  about  50  volts.  Since 


DYNAMOS  297 

an  arc  light  usually  requires  a  current  of  10  amperes,  such  a  dynamo 
must  furnish  10  amperes  at  2500  volts.  The  power  is  therefore 
10  x  2500  —  25,000  watts.  The  dynamo  must  therefore  have  an.  activity 
of  25  kilowatts,  or  about  33.5  horse  power. 

369.  The  electric  motor.    In  construction  the  electric  motor 
differs  in  no  essential  respect  from  the  dynamo.    To  analyze 
the  operation  as  a  motor  of  such  a  machine  as  that  shown  in 
Fig.  315,  suppose  a  current  from  an  outside  source  is  first  sent 
around  the  coils  of  the  field  magnets  and  then  into  the  arma- 
ture at  b'.    Here  it  will  divide  and  flow  through  all  the  con- 
ductors on  the  left  half  of  the  ring  in  one  direction,  and  through 
all  those  on  the  right  half  in  the  opposite  direction.    Hence, 
in  accordance  with  the  motor  rule,  all  the  conductors  on  the 
left  side  are  urged  upward  by  the  influence  of  the  field,  and 
all  those  on  the  right  side  are  urged  downward.    The  armature 
will  therefore  begin  to  rotate,  and  this  rotation  will  continue 
so  long  as  the  current  is  sent  in  at  bf  and  out  at  b.    For  as  fast 
as  coils  pass  either  b  or  6',  the  direction  of  the  current  flowing 
through  them  changes,  and  therefore  the  direction  of  the  force 
acting  on  them  changes.    The  left  half  is  therefore  always 
urged  up  and  the  right  half  down.    The  greater  the  strength  of 
the  current,  the  greater  the  force  acting  to  produce  rotation. 

If  the  armature  is  of  the  drum  type  (Fig/317),  the  conditions 
are  not  essentially  different.  For,  as  may  be  seen  by  following 
out  the  windings,  the  current  entering  at  b'  will  flow  through 
all  the  conductors  in  the  left  half  in  one  direction  and  through 
those  on  the  right  half  in  the  opposite  direction.  The  com- 
mutator keeps  these  conditions  always  fulfilled.  The  analysis 
of  the  operation  of  a  four-pole  dynamo  (Fig.  316)  as  a  motor 
is  equally  simple. 

370.  Street-car  motors.    Electric  street  cars  are  nearly  all  operated  by 
direct-current  series-wound  motors  placed  under  the  cars  and  attached 
by  gears  to  the  axles.    Fig.  323  shows  a  typical  four-pole  street-car 
motor.    The  two  upper  field  poles  are  raised  with  the  case  when  the 


298 


INDUCED  CURRENTS 


motor  is  opened  for  inspection,  as  in  the  figure.  The  current  is  generally 
supplied  by  compound-wound  dynamos  which  maintain  a  constant 
potential  of  about  500 
volts  between  the  trol- 
ley, or  third  rail,  and 
the  track  which  is  used 
as  the  return  circuit. 
The  cars  are  always 
operated  in  parallel,  as 
shown  in  Fig.  324.  In 
a  few  instances  street 
cars  are  operated  upon 
alternating,  instead  of 
upon  direct-current,  cir- 
cuits. In  such  cases  the 
motors  are  essentially  the  same  as  direct-current  series-wound  motors ; 
for  since  in  such  a  machine  the  current  must  reverse  in  the  field 


FIG.  323.    Hallway  motor,  upper  field  raised 


Trolley  Wire  or  3rd.  Rail 


nnnnnnnni  ] 

1 

DODOD 

uencraiorr 
at  Power  \     > 
Station    ^\^ 

L—  'l  )   1  )                  O    LJ  ^ 

"  —  '  C  )            (.  )  '  —  ' 

Track 
FIG.  324.    Street-car  circuit 


magnets  at  the  same  time  that  it  reverses  in  the  armature,  it  will  be 
seen  that  the  armature  is  always  impelled  to  rotate  in  one  direction, 
whether  it  is  supplied  with  a  direct  or  with  an  alternating  current. 

371.  Back  E.M.F.  in  motors.  When  an  armature  is  set  into 
rotation  by  sending  a  current  from  some  outside  source  through 
it,  its  coils  move  through  a  magnetic  field  as  truly  as  if  the 
rotation  were  produced  by  a  steam  engine,  as  is  the  case  in 
running  a  dynamo.  An  induced  E.M.F.  is  therefore  set  up 
by  this  rotation.  In  other  words,  while  the  machine  is  acting 
as  a  motor  it  is  also  acting  as  a  dynamo.  The  direction  of  the 
induced  E.M.F.  due  to  this  dynamo  effect  will  be  seen,  from 
Lenz's  law  or  from  a  consideration  of  the  dynamo  and  motor 
rules,  to  be  opposite  to  the  outside  P.D.,  which  is  causing 


DYNAMOS  299 

current  to  pass  through  the  motor.  The  faster  the  motor  rotates, 
the  faster  the  lines  of  force  are  cut,  and  hence  the  greater  the 
value  of  this  so-called  back  E.M.F.  If  the  motor  were  doing  no 
work,  the  speed  of  rotation  would  increase  until  the  back  E.M.F. 
reduced  the  current  to  a  value  simply  sufficient  to  overcome 
friction.  It  will  be  seen,  therefore,  that  in  general  the  faster  the 
motor  goes,  the  less  the  current  which  passes  through  its  arma- 
ture, for  this  current  is  always  due  to  the  difference  between 
the  P.D.  applied  at  the  brushes  —  500  volts  in  the  case  of 
trolley  cars  —  and  the  back  E.M.F.  When  the  motor  is  start- 
ing, the  back  E.M.F.  is  zero ;  and  hence,  if  the  full  500  volts 
were  applied  to  the  brushes,  the  current  sent  through  would 
be  so  large  as  to  ruin  the  armature  through  overheating.  To 
prevent  this  each  car  is  furnished  with  a  "  starting  box,"  which 
consists  of  resistance  coils  which  the  motorman  throws  into 
series  with  the  motor  on  starting,  and  throws  out  again 
gradually  as  the  speed  increases  and  the  back  E.M.F.  conse- 
quently rises.* 

QUESTIONS  AND  PROBLEMS 

1.  Explain  how  an  alternating  current  in  the  armature  is  trans- 
formed into  a  unidirectional  current  in  the  external  circuit. 

2.  Two  successive  coils  on  the  armature  of  a  multipolar  alternator 
are  cutting  lines  of  force  which  run  in  opposite  directions.    How  does 
it  happen  that  the  currents  generated  flow  through  the  wires  in  the 
same  direction  ?    (Fig.  312.) 

3.  A  multipolar  alternator  has  20  poles  and  rotates  200  times  per  min- 
ute.   How  many  alternations  per  second  will  be  produced  in  the  circuit  ? 

4.  With  the  aid  of  the  dynamo  rule  explain  why,  in  Fig.  316,  the 
current  in  the  conductors  under  the  south  poles  is  moving  toward  the 
observer,  and  that  in  the  conductors  under  the  north  poles  away  from 
the  observer.    Explain  in  a  similar  way  the  directions  of  the  arrows  in 
Figs.  315  and  317. 

5.  Explain  why  the  brushes  in  Fig.  316  touch  the  commutator  in 
the  positions  shown  rather  than  at  some  other  points. 

*  This  discussion  should  be  followed  by  a  laboratory  experiment  on  the  study 
of  a  small  electric  motor  or  dynamo.  See,  for  example,  Experiment  No.  37  of  the 
authors'  manual. 


300  ,  INDUCED  CURRENTS 

•  6.  If  a  direct-current  machine  of  the  same  general  type  as  that 
shown  in  Fig.  316  had  twelve  poles,  how  many  brushes  would  be  needed 
on  the  commutator  ? 

7.  If  a  current  is  sent  into  the  armature  of  Fig.  315  at  b',  and  taken 
out  at  b,  which  way  will  the  armature  revolve  ? 

8.  When  an  electric  fan  is  first  started, the  current  through  it  is  much 
greater  than  it  is  after  the  fan  has  attained  its  normal  speed.    Why? 

9.  If  in  the  machine  of  Fig.  316  a  current  is  sent  in  on  the  wire 
marked  + ,  what  will  be  the  direction  of  rotation  ? 

10.  Would  an  armature  wound  on  a  wooden  core  be  as  effective  as 
one  made  of  the  same  number  of  turns  wound  on  an  iron  core  ? 

11.  Will  it  take  more  work -to  rotate  a  dynamo  armature  when  the 
circuit  is  closed  than  when  it  is  open  ?   Why  ? 

12.  Show  that  if  the  reverse  of  Lenz's  law  were  true,  a  motor  once 
started  would  run  of  itself  and  do  work ;  that  is,  it  would  furnish  a  case 
of  perpetual  motion. 

13.  If  a  series-wound  dynamo  is  running  at  a  constant  speed,  what 
effect  will  be  produced  on  the  strength  of  the  field  magnets  by  dimin- 
ishing the  external  resistance  and  thus  increasing  the  current?    What 
will  be  the  effect  on  the  E.M.F.  ?    (Remember  that  the  whole  current 
goes  around  the  field  magnets.) 

14.  If  a  shunt-wound  dynamo  is  run  at  constant  speed,  what  effect 
will  be  produced  on  the  strength  of  the  field  magnets  by  reducing  the 
external  resistance  ?   What  effect  will  this  have  on  the  E.M.F.  ? 

15.  In  an  incandescent-lighting  system  the  lamps  are  connected  in 
parallel  across  the  mains.  Every  lamp  which  is  turned  on,  then,  dimin- 
ishes the  external  resistance.    Explain  from  a  consideration  of  Problems 
13  and  14  why  a  compound-wound  dynamo  keeps  the  P.D.  between  the 
mains  constant. 

16.  Explain  why  a  series-wound  motor  can  run  either  on  a  direct  or 
an  alternating  circuit. 

17.  If  the  pressure  applied  at  the  terminals  of  a  motor  is  500  volts, 
and  the  back  pressure,  when  running  at  full  speed,  is  450  volts,  what  is 
the  current  flowing  through  the  armature,  its  resistance  being  10  ohms  ? 

18.  Single  dynamos  often  operate  as  many  as  10,000  incandescent 
lamps  at  110  volts.  If  these  lamps  are  all  arranged  in  parallel  and  each 
requires  a  current  of  .5  ampere,  what  is  the  total  current  furnished  by 
the  dynamo  ?   AVhat  is  the  activity  of  the  machine  in  kilowatts  and  in 
horse  power  ? 

19.  How 'many  110-volt  lamps  like  those  of  Problem  18  can  be 
lighted  by  a  12,000-kilowatt  generator  ? 

20.  Why  does  it  take  twice  as  much  work  to  keep  a  dynamo  running 
when  1000  lights  are  on  the  circuit  as  when  only  500  are  turned  on  ? 


INDUCTION  COIL  AND  TRANSFORMER         301 

PRINCIPLE  OF  THE  INDUCTION  COIL  AND  TRANSFORMER 

372.  Currents  induced  by  varying  the  strength  of  a  magnetic 

field.  Let  about  500  turns  of  No.  28  copper  wire  be  wound  around  one 
end  of  an  iron  core,  as  in  Fig.  325,  and  connected  to  the  circuit  of  a 
galvanometer.  Let  about  500  more  turns  be  wrapped  about  another 
portion  of  the  core  and  connected  into  the  circuit  of  two  dry  cells.  When 
the  key  K  is  closed  the  deflection  of  the  galvanometer  will  indicate  that 
a  temporary  current  has  been  induced  in  one  direction  through  the  coil  s, 
and  when  it  is  opened  an  equal  but  opposite  deflection  will  indicate  an 
equal  current  flowing  in  the  opposite  direction. 

The  experiment  illustrates  the  principle  of  the  induction 
coil  and  the  transformer.    The  coil  p,  which  is  connected  to  the 
source  of  the  current, 
is  called  the  primary      **T^\>^ 

coil,  and  the  coil  s,  in  ^v\  /" * — ~Hl-\ 

which  the  currents  are     nof]!  \^£»=m^ ) 

induced,  is  called  the     \Lj) 

secondary   coil.    Caus- 

FIG.  325.   Induction  of  current  by  magnetizing 

ing    lines    OI   lorce   to  an(j  demagnetizing  an  iron  core 

spring  into   existence 

inside  of  s  —  in  other  words,  magnetizing  the  space  inside  of  s  — 
has  caused  an  induced  current  to  flow  in  s ;  and  demagnetizing 
the  space  inside  of  s  has  also  induced  a  current  in  s  in  accord- 
ance with  the  general  principle  stated  in  §  358,  that  any  change 
in  the  number  of  magnetic  lines  of  force  which  thread  through  a 
coil  induces  a  current  in  the  coil.  We  may  think  of  the  lines 
as  always  existing  as  closed  loops  (see  Fig.  285,  p.  274)  which 
collapse  upon  demagnetization  to  mere  double  lines  at  the  axis 
of  the  coil.  Upon  magnetization  one  of  these  two  lines  springs 
out,  cutting  the  encircling  conductors  and  inducing  a  current. 

373.  Direction  of  the  induced  current.    Lenz's  law,  which, 
it  will  be  remembered,  followed  from  the  principle  of  conser- 
vation of  energy,  enables  us  to  predict  at  once  the  direction 
of  the  induced  currents  in  the  above  experiments ;   and  an 


302  INDUCED  CURRENTS 

observation  of  the  deflections  of  the  galvanometer  enables  us  to 
verify  the  correctness  of  the  predictions.  Consider  first  the  case 
in  which  the  primary  circuit  is  made  and  the  core  thus  magnet- 
ized. According  to  Lenz's  law,  the  current  induced  in  the  sec- 
ondary circuit  must  be  in  such  a  direction  as  to  oppose  the  change 
which  is  being  produced  by  the  primary  current,  that  is,  in  such 
a  direction  as  to  tend  to  magnetize  the  core  oppositely  to  the 
direction  in  which  it  is  being  magnetized  by  the  primary.  This 
means,  of  course,  that  the  induced  current  in  the  secondary 
must  encircle  the  core  in  a  direction  opposite  to  the  direction 
in  which  the  primary  current  encircles  it.  We  learn,  therefore, 
that  on  making  the  current  in  the  primary  the  current  induced 
in  the  secondary  is  opposite  in  direction  to  that  in  the  primary. 

When  the  current  in  the  primary  is  broken,  the  magnetic 
field  created  by  the  primary  tends  to  die  out.  Hence,  by  Lenz's 
law,  the  current  induced  in  the  secondary  must  be  in  such  a 
direction  as  to  tend  to  oppose  this  process  of  demagnetization, 
that  is,  in  such  a  direction  as  to  magnetize  the  core  in  the  same 
direction  in  which  it  is  magnetized  by  the  decaying  current 
in  the  primary.  Therefore,  at  break  the  current  induced  in  the 
secondary  is  in  the  same  direction  as  that  in  the  primary. 

374.  E.M.F.  of  the  secondary.    If  half  of  the  500  turns  of 
the  secondary  s  (Fig.  325)  are  unwrapped,  the  deflection  will 
be  found  to  be  just  half  as  great  as  before.   Since  the  resistance 
of  the  circuit  has  not  been  changed,  we  learn  from  this  that 
the  E.M.F.  of  the  secondary  is  proportional  to  the  number  of 
turns  of  wire  upon  it  '• —  a  result  which  followed  also  from 
§  361.    If,  then,  we  wish  to  develop  a  very  high  E.M.F.  in 
the  secondary,  we  have  only  to-make  it  of  a  very  large  number 
of  turns  of  fine  wire. 

375.  Self-induction.     If   in  the   experiment  illustrated   in 
Fig.  325  the  coil  s  had  been  made  a  part  of  the  same  circuit  as 
£>,  the  E.M.F.'s  induced  in  it  by  the  changes  in  the  magnetism 
of  the  core  would  of  course  have  been  just  the  same  .as  above. 


INDUCTION  COIL  AND  TRANSFORMER         303 

In  other  words,  when  a  current  starts  in  a  coil  the  magnetic 
field  which  it  itself  produces  tends  to  induce  a  current  oppo- 
site in  direction  to  that  of  the  starting  current,  that  is,  tends 
to  oppose  the  starting  of  the  current;  and  when  a  current 
in  a  coil  stops,  the  collapse  of  its  own  magnetic  field  tends  to 
induce  a  current  in  the  same  direction  as  that  of  the  stopping 
current,  that  is,  tends  to  oppose  the  stopping  of  the  current. 
This  means  merely  that  a  current  in  a  coil  acts  as  though  it  had 
inertia,  and  opposes  any  attempt  to  start  or  stop  it.  This  inertia- 
like  effect  of  a  coil  upon  itself  is  called  self-induction. 

Let  a  few  dry  cells  be  inserted  into  a  circuit  containing  a  coil  of  a  . 
large  number  of  turns  of  wire,  the  circuit  being  closed  at  some  point  by  ! 
touching  two  bare  copper  wires  together.    Holding  the  bare  wire  in  the   ! 
fingers,  break  the  circuit  between  the  hands  and  observe  the  shock  due  to 
the  current  which  the  E.M.F.  of  self-induction  sends  through  your  body. 
Without  the  coil  in  circuit  you  will  obtain  no  such  shock,  though  the    ' 
current  stopped  when  you  break  the  circuit  will  be  many  times-larger. 

The  spark  coil  on  an  automobile  is  a  good  illustration  of  a  device  for^ 
producing  a  spark  due  to  self-induction. 

%*%'' 

376.  The   induction   coil.    The  induction    coil,   as   usually 

made   (Fig.  326),   consists  of  a  soft  iron  core  (7,  composed 

of  a  bundle  of  soft  iron  wires  ;  a 
primary  coil  p  wrapped  around 


FIG.  326.    Induction  coil 


this  core,  and  consisting  of,  say,  200  turns  of  coarse  copper  wire 
(for  example,  No.  16),  which  is  connected  into  the  circuit  of 
a  battery  through  the  contact  point  at  the  end  of  the  screw 
d ;  a  secondary  coil  *  surrounding  the  primary  in  the  manner 


304  .  INDUCED  CURRENTS 

indicated  in  the  diagram,  and  consisting  generally  of  between 
30,000  and  1,000,000  turns  of  No.  36  copper  wire,  the  termi- 
nals of  which  are  the  points  t  and  t' ;  and  a  hammer  6,  or 
other  automatic  arrangement  for  making  and  breaking  the 
circuit  of  the  primary. 

Let  the  hammer  I  be  held  away  from  the  opposite  contact  point  by 
means  of  the  finger,  then  touched  to  this  point,  then  pulled  quickly  away. 
A  spark  will  be  found  to  j>ass  between  t  and  t'  at  break  only  —  never  at  make. 
This  is  because,  on  account  of  the  opposing  influence  at  make  of  self- 
induction  in  the  primary,  the  magnetic  field  about  the  primary  rises 
very  gradually  to  its  full  strength,  and  hence  its  lines  pass  into  the  sec- 
ondary coil  comparatively  slowly.  At  break,  however,  by  separating  the 
contact  points  very  quickly  we  can  make  the  current  in  the  primary  fall 
to  zero  in  an  exceedingly  short  time,  perhaps  not  more  than  .00001 
second ;  that  is,  we  can  make  all  of  its  lines  pass  out  of  the  coil  in  this 
time.  Hence  the  rate  at  which  lines  thread  through  or  cut  the  secondary 
is  perhaps  10,000  times  as  great  at  break  as  at  make,  and  therefore  the 
E.M.F.  is  also  something  like  10,000  times  as  great.  In  the  normal  use 
of  the  coil,  the  circuit  of  the  primary  is  automatically  made  and  broken 
at  b  by  means  of  the  magnet  and  the  spring  r,  precisely  as  in  the  case  of 
the  electric  bell.  Let  the  student  analyze  this  part  of  the  coil  for  him- 
self. The  condenser,  shown  in  the  diagram,  with  its  two  sets  of  plates 
connected  to  the  conductors  on  either  side  of  the  spark  gap  between  r 
and  d,  is  not  an  essential  part  of  a  coil,  but  when  it  is  introduced  it  is 
found  that  the  length  of  the  spark  which  can  be  sent  across  between  t 
and  if  is  considerably  increased.  The  reason  is  as  follows :  When  the 
circuit  is  broken  at  b  the  inertia,  that  is,  the  self-induction,  of  the  pri- 
mary current,  tends  to  make  a  spark  jump  across  from  d  to  b ;  and  if 
this  happens,  the  current  continues  to  flow  through  this  spark  (or  arc) 
until  the  terminals  have  become  separated  through  a  considerable  dis- 
tance. This  makes  the  current  die  down  gradually  instead  of  suddenly, 
as  it  ought  to  do  to  produce  a  high  E.M.F.  But  when  a  condenser  is  in- 
serted, as  soon  as  b  begins  to  leave  d  the  current  begins  to  flow  into  the 
condenser,  and  this  gives  the  hammer  time  to  get  so  far  away  from 
d  that  an  arc  cannot  be  formed.  This  means  a  sudden  break  and  a 
high  E.M.F.  Since  a  spark  passes  between  t  and  if  only  at  break,  it 
must  always  pass  in  the  same  direction.  Coils  which  give  24-inch  sparks 
(perhaps  500,000  volts)  are  not  uncommon.  Such  coils  usually  have 
hundreds  of  miles  of  wire  upon  their  secondaries. 


INDUCTION  COIL  AND  TRANSFOKMEK         305 


377.  Laminated  cores  ;  Foucault  currents.    The  core  of  an  induction 
coil  should  always  be  made  of  a  bundle  of  soft  iron  wires  insulated 
from  one  another  by  means  of  shellac  or  varnish 

(see  Fig.  327)  ;  for  whenever  a  current  is  started 
or  stopped  in  the  primary  p  of  a  coil  furnished 
with  a  solid  iron  core  (see  Fig.  328),  the  change  in 
the  magnetic  field  of  the  primary  induces  a  cur- 
rent in  the  conducting  core  C  for  the  same  reason 
that  it  induces  one  in  the  secondary  s.  This  cur-  -^  007  r<ore  of  :n 
rent  flows  around  the  body  of  the  core  in  the  sulated  iron  wires 
same  direction  as  the*  induced  current  in  the 

secondary,  that  is,  in  the  direction  of  the  arrows.    The  only  effect  of 
these  so-called  eddy  or  Foucault  currents  is  to  heat  the  core.    This  is 
obviously  a  waste  of  energy.    If  we  can  prevent  the 
appearance    of  these    currents,    all    of    the  energy 
-which  they  would  waste  in  heating  the  core  may 
be  made  to  appear  in  the  current  of  the  secondary. 
The  core  is  therefore  built  of  varnished  iron  wires, 
which  run  parallel  to  the  axis  of  the  coil,  that  is, 
perpendicular  to  the  direction  in  which  the  cur- 
rents would  be  induced.  The  induced  E.M.F.  there- 
fore finds  no  closed  circuits  in  which  to  set  up  a 
current  (Fig.  327).    It  is  for  the  same  reason  that 
the  iron  cores  of  dynamo  and  motor  armatures,  instead  of  being  solid,  con- 
sist of  iron  disks  placed  side  by  side,  as  shown  in  Fig.  329,  and  insulated 
from   one  another    by  films    of 
oxide.    A  core  of  this  kind  is 
called  a  laminated  core.    It  will 
be  seen  that  in  all  such  cores 
the  spaces  or  slots  between  the 
laminse  must  run  at  right  angles 
to  the  direction  of  the  induced 
E.M.F.,  that  is,  perpendicular  to 
the  conductors   upon  the  core. 

378.  The  transformer.    The   commercial  transformer  is  a 
modified  form  of  the  induction  coil.    The  chief  difference  is 
that  the  core  R  (Fig.  330),  instead  of  being  straight,  is  bent 
into  the  form  of  a  ring,  or  is  given  some  other  shape  such 
that    the    magnetic    lines    of   force    have   a   continuous   iron 


FIG.  328.  Diagram 
showing  eddy  cur- 
rents in  solid  core 


FIG.  329.    Laminated  drum-armature 

core   with    commutator,    showing    one 

coil  wound  on  the  core 


306 


INDUCED  CURRENTS 


path,  instead  of  being  obliged  to  push  out  into  the  air,  as  in 
the  induction  coil.  Furthermore,  it  is  always  an  alternating 
instead  of  an  intermittent  current  which 
is  sent  through  the  primary  A.  Send- 
ing such  a  current  through  A  is  equiva- 
lent to  magnetizing  the  core  first  in  one 
direction,  then  demagnetizing  it,  then 
magnetizing  it  in  the  opposite  direc- 
tion, etc.  The  results  of  these  changes 
in  the  magnetism  of  the  core  is  of 

course    an  induced  alternating  current  in  the  secondary  7?. 
379.  The  use  of  the  transformer.    The  use  of  the  transformer 


FIG.  330. 

transformer 


R 

Diagram  of 


is  to   convert  an 


alternating 


current   from 


one 


voltage   to 


another  which,  for  some  reason,  is  found  to  be  more  convenient. 
For  example,  in  electric  lighting  where  an  alternating  current 
is  used,  the  E.M.F.  gen- 
erated by  the  dynamo 
is  usually  either  1100 
2200  volts,  a  volt- 


Main  Conditctor 


tynamo 


FIG.  331.  Alternating  current  lighting  circuit 
with  transformers 


or 

age  too  high  to  be  in- 
troduced safely  into 
private  houses.  Hence 
transformers  are  con- 
nected across  the  main 
conductors  in  the  man- 
ner shown  in  Fig.  331. 
The  current  which  passes  into  the  houses  to  supply  the  lamps 
does  not  come  directly  from  the  dynamo.  It  is  an  induced 
current  generated  in  the  transformer. 

380.  Pressure  in  primary  and  secondary.  If  there  are  a  few 
turns  in  the  primary  and  a  large  number  in  the  secondary,  the 
transformer  is  called  a  step-up  transformer,  because  the  P.D. 
produced  at  the  terminals  of  the  secondary  is  greater  than  that 
applied  at  the  terminals  of  the  primary.  Thus  an  induction 


INDUCTION  COIL  AND  TRANSFORMER         SOT 


coil  is  a  step-up  transformer.  In  electric  lighting,  however, 
transformers  are  mostly  of  the  step-down  type ;  that  is,  a  high 
P.D.,  say,  2200  volts,  is  applied  at  the  terminal  of  the  primary, 
and  a  lower  P.D.,  say,  110  volts,  is  obtained  at  the  terminals 
of  the  secondary.  In  such  a  transformer  the  primary  will  have 
twenty  times  as  many  turns  as  the  secondary.  In  general,  the 
ratio  between  the  voltages  at  the  terminals  of  the  primary  and 
secondary  is  the  ratio  of  the  number  of  turns  of  wire  upon  the  two. 
381.  Efficiency  of  the  transformer.  In  a  perfect  transformer 
the  efficiency  would  be  unity.  This  means  that  the  electrical 
energy  put  into  the  primary,  that  is,  the  volts  applied  to  its 
terminals  times  the  amperes  flowing  through  it,  would  be  ex- 
actly equal  to  the  energy  taken  out  in  the  secondary,  that  is, 
the  volts  generated  in  it  times  the  strength  of  the  induced 
current ;  and,  in  fact,  in  actual  transformers  the  latter  prod- 
uct is  often  more  than  97%  of  the  former;  that  is,  there  is 
less  than  3%  loss  of  energy  in  the  transformation.  This  lost 
energy  appears  as  heat  in  the  transformer.  This  transfer, 
which  goes  on  in  a  big  transformer,  of  huge  quantities  of 
power  from  one  circuit  to  another  entirely  independent  cir- 
cuit, without  noise  or  motion  of  any  sort  and  almost  without 

loss,  is  one  of  the  most  wonderful  phenomena 

of  modern  industrial  life.  > 


FIG.  332.    Commer- 
cial transformer 


FIG.  333.  Cross  section 
of  transformer,  show- 
ing shape  of  magnetic 
field 


FIG.  334.     Trans- 
former case 


382.  Commercial  transformers.  Fig.  332  illustrates  a  common  type  of 
transformer  used  in  electric  lighting.  The  core  is  built  up  of  sheet-iron 
laminae  about  \  millimeter  thick.  Fig.  333  shows  a  section  of  the  same 

t 


308 


INDUCED  CURRENTS 


'sformer 


FIG.  335.    Transformer  on  electric- 
light  pole 


transformer.    The  closed  magnetic  circuit  of  the  core  is  indicated  by  the 

arrows.    The  primary  and  the  two  secondaries,  which  can  furnish  either 

52  or  104  volts,  are  indicated  by  the 

letters  jo,  Sv  and  52.  Fig.  334  is  the 

case  in  which    the    transformer  is 

placed.     Such   cases  may   be   seen 

attached  to  poles  outside  of  houses 

wherever  alternating    currents    are 

used  for  electric-lighting  (Fig.  335). 

383.  Electrical  transmission  of 
power.  Since  the  electrical  energy 
produced  by  a  dynamo  is  equal  to  the 
product  of  the  E.M.F.  generated  by 
the  current  furnished,  it  is  evident 
that  in  order  to  transmit  from  one 
point  to  another  a  given  number  of 
watts,  say,  10,000,  it  is  possible  to 
have  either  an  E.M.F.  of  100  volts 
and  a  current  of  100  amperes,  or  an 
E.M.F.  of  1000  volts  and  a  current  of 
10  amperes.  In  the  two  cases,  how- 
ever, the  loss  of  energy  in  the  wire  which  carries  the  current  from  the 
place  where  it  is  generated  to  the  place  where  it  is  used  will  be  widely 
different.  If  R  represents  the  resistance  of  this  transmitting  wire,  the  so- 
called  **  line,"  and  C  the  current  flowing  through  it,  we  have  seen  in  §  351 
that  the  heat  developed  in  it  will  be  proportional  to  C2R.  Hence  the 
energy  wasted  in  heating  the  line  will  be  but  -j-^-j  as  much  in  the  case  of 
the  high-voltage,  10-ampere  current  as  in  the  case  of  the  lower-voltage, 
100-ampere  current.  Hence,  for  long-distance  transmission,  where  line 
losses  are  considerable,  it  is  important  to  use  the  highest  possible  voltages. 

On  account  of  the  difficulty  of  insulating  the  commutator  segments 
from  one  another,  voltages  higher  than  700  or  800  cannot  be  obtained 
with  direct-current  dynamos  of  the  kind  which  have  been  described. 
With  alternators,  however,  the  difficulties  of  insulation  are  very  much 
less  on  account  of  the  absence  of  a  commutator.  The  large  10,000-horse- 
power  alternating-current  dynamos  on  the  Canadian  side  of  Niagara 
Falls  generate  directly  12,000  volts.  This  is  the  highest  voltage  thus 
far  produced  by  generators.  In  all  cases  where  these  high  pressures  are 
employed  they  are  transformed  down  at  the  receiving  end  of  the  line 
to  a  safe  and  convenient  voltage  (from  50  to  500  volts)  by  means  of 
step-down  transformers. 


INDUCTION  COIL  AND  TRANSFORMER 


309 


I V/WVWWV\M ' 

Transformer 

i WWWW 1, 

A.C.  Supply 


It  will  be  seen  from  the  above  facts  that  only  alternating  currents 
are  suitable  for  long-distance  transmission.  Plants  are  now  in  operation 
which  transmit  power  as  far  as  150  miles  and  use  pressures  as  high  as 
100,000  volts.  In  all  such  cases  step-up  transformers,  situated  at  the 
power  house,  transfer  the  electrical  energy  developed  by  the  generator 
to  the  line,  and  step-down  transformers,  situated  at  the  receiving  end, 
transfer  it  to  the  motors,  or  lamps,  which  are  to  be  supplied.  The  gen- 
erators used  on  the  American  side  of  Niagara  Falls  produce  a  pressure 
of  2300  volts.  For  transmission  to  Buffalo,  20  miles  away,  this  is  trans- 
formed up  to  22,000  volts.  At  Buffalo  it  is  transformed  down  to  the 
voltages  suitable  for  operating  the  street  cars,  lights,  and  factories  of 
the  city.  On  the  Canadian  side  the  generators  produce  currents  at 
12,000  volts,  as  stated,  and  these  are  transformed  up,  for  long-distance 
transmission,  to  22,000,  40,000,  and  60,000  volts. 

384.  The  mercury-arc  rectifier.  The  mercury-arc  rectifier  is  a  recently 
developed  instrument  for  changing  an  alternating  to  a  direct  current. 
It  consists  of  two  graphite  anodes  A 
and  A'  (Fig.  336),  and  a  mercury 
cathode  B  in  an  exhausted  bulb.  It  is 
found  that  a  current  will  pass  through 
such  a  bulb  when  t*he  carbon  is  made 
the  positive  and  the  mercury  the  nega- 
tive electrode,  but  not  in  the  reverse 
direction.  When,  then,  an  alternating 
E.M.F.  is  applied  at  H  and  G,  the  cur- 
rent passes  through  the  circuit  first  in 
the  direction  indicated  by  the  plain 
arrows,  and  then,  as  the  E.M.F.  re- 
verses, in  the  direction  indicated  by 
the  circled  arrows.  It  will  be  seen  that 
it  always  passes  in  the  same  direction 
through  the  storage  batteries  J  which 
are  to  be  charged.  Were  it  not  for 
the  large  coils  EF  the  transformer 
would  be  short-circuited  through  PDQ 
and  no  current  would  flow  through 
the  path  MED  or  NBD.  But  the 
self-inductions  of  E  and  F  are  so  large  that  most  of  the  current  flowing 
from  M  to  D  or  TV  to  D  is  forced  over  the  path  MBD  or  NBD.  The 
extra  mercury  electrode  C  and  the  resistance  coil  0  are  merely  used  for 
starting  the  rectifier.  This  is  done  by  til  ting  it  until  the  mercury  in  B  and 


FIG.  336.   The  mercury-arc 
rectifier 


310 


INDUCED  CURRENTS 


C  makes  contact,  then  righting  and  thus  breaking  this  contact  and 
forming  a  temporary  arc.  This  puts  the  mercury  vapor  into  condition 
to  cause  the  rectifier  to  function  as  described. 

385.  The  simple  telephone.  The  telephone  was  invented  in 
1875  by  Alexander  Graham  Bell,  of  Washington,  and  Elisha 
Gray,  of  Chicago.  In  its  simplest  form  it  consists,  at  each 
end,  of  a  permanent  bar  mag- 
net A  (Fig.  337)  surrounded 
by  a  coil  of  fine  wire  B,  in 
series  with  the  line,  and  an 


Jron  disk  or  diaphragm  E 
mounted  close  to  one  end  of 
the  magnet.  When  a  sound 


FIG.  337.   The  simple  telephone 


is  made  in  front  of  the  diaphragm,  the  vibrations  produced  by 
th©  sounding  body  are  transmitted  by  the  air  to  the  diaphragm, 
thus  causing  the  latter  to  vibrate  back  and  forth  in  front  of 
the  magnet.  These  vibrations  of  the  diaphragm  produce  slight 
backward  and  forward  movements  of  the  lines  of  force  which 
pass  into  the  disk  from  the  magnet  in  the  manner  shown  in 
Fig.  338.  Some  of  these  lines  of  force,  therefore,  cut  across 
the  coil  B,  first  in  one  direction  and  then  in  the  other,  and  in 
so  doing  induce  currents  in  it.  (These 
induced  currents  are  transmitted  by 
the  line  to  the  receiving  station,  where 
those  in  one  direction  pass  around  B'  in 
such  a  way  as  to  increase  the  strength 
of  the  magnet  A',  and  thus  increase 
the  pull  which  it  exerts  upon  E\  while 
the  opposite  currents  pass  around  B1 

in  the  opposite  direction,  and  therefore  weaken  the  magnet  A' 
and  diminish  its  pull  upon  E'.  When,  therefore,  E  moves  in 
one  direction,  E'  also  moves  in  one  direction,  and  when  E 
reverses  its  motion,  the  direction  of  E'  is  also  reversed.  In 
other  words,  the  induced  currents,  transmitted  by  the  line, 


\i»'&r__  _ 


FIG.  338.     Magnetic  field 
about  a  telephone  receiver 


©  Clinediust 

ALEXANDER  GRAHAM  BELL, 
WASHINGTON,  D.C. 

Inventor  of  the  telephone,  1875 


©  Underwood  &  Underwood 

THOMAS  A.  EDISON,  ORANGE, 
NEW  JERSEY 

Inventor  of  the  phonograph,  the  incan- 
descent lamp,  etc. 


GUGLIELMO  MARCONI  (ITALY)  ORVILLE  WRIGHT,  DAYTON,  OHIO 

Inventor  of  commercial  wireless  Inventor,  with  his  brother  Wilhur,  of 

telegraphy  the  aeroplane 

A  GROUP  OF  MODERN  INVENTORS 


INDUCTION  COIL  AND  TRANSFORMER         311 


force  E'  to  reproduce  the  motions  of  E.  E'  therefore  sends  out 
sound  waves  exactly  like  those  which  fell  upon  E.  In  exactly 
the  same  way  a  sound  made  in  front  of  E'  is  reproduced  at  E. 
Telephones  of  this  simple  type  will  work  satisfactorily  for  a 
distance  of  several  miles.  This  simple  form  of  instrument  is 
still  used  at  the  receiving  end  of  the 
modern  telephone,  the  only  innovation 
which  has  been  introduced  consisting 
in  the  substitution  of  a  U-shaped  mag- 
net for  the  bar  magnet.  The  instru- 
ment used  at  the  transmitting  end  has, 
however,  been  changed,  as  explained  in 
the  next  paragraph,  and  the  circuit  is  now  completed  through 
a  return  wire  instead  of  through  the  earth.  A  modern  tele- 
phone receiver  is  shown  in  Fig.  339.  G  is  the  mouthpiece, 
Ethe  diaphragm,  A  the  U-shaped  magnet,  and  B  the  coils,  con- 
sisting of  many  turns  of  fine  wire,  and  having  soft,  iron 


FIG.  339.   The  modern 
receiver 


386.  The  modern  transmitter.  To  increase  the  distance  at  which  tele- 
phoning may  be  done,  it  is  necessary  to  increase  the  strength  of  the 
induced  currents.  This  is  done  in  the  modern  transmitter  by  replacing 
the  magnet  and  coil  by  an  arrangement  which  is  essentially  an  induc- 
tion coil,  the  current  in  the  primary  of  which  is  caused  to  vary  by  the 

Receiver  Receiver 


B  B 

FIG.  340.    The  telephone  circuit  (local-battery  system) 

motion  of  the  diaphragm.  This  is  accomplished  as  follows :  The  cur- 
rent from  the  battery  B  (Fig.  340)  is  led  first  to  the  back  of  the 
diaphragm  E,  whence  it  passes  through  a  little  chamber  C,  filled  with 
granular  carbon,  to  the  conducting  back  d  of  the  transmitter,  and  thence 
through  the  primary  p  of  the  induction  coil,  and  back  to  the  battery 


312 


INDUCED  CURRENTS 


As  the  diaphragm  vibrates  it  varies  the  pressure  upon  the  many  contact 
points  of  the  granular  carbon  through  which  the  primary  current  flows. 
This  produces  considerable  variation  in  the  re- 
sistance of  the  primary  ..circuit,  so  that  as  the 
diaphragm  moves  forward,  that  is,  toward  the 
carbon,  a  comparatively  large  current  flows 
through  p,  and,  as  it  moves  back,  a  much 
smaller  current.  These  changes  in  the  current 
strength  in  the  primary  p  produce  changes  in 
the  magnetism  of  the  sofi>iron  core  of  the  in- 
duction coil.  Currents  are  therefore  induced 
in  the  secondary  s  of  the  induction  coil,  and 
these  currents  pass  over  the  line  and  affect 
the  receiver  at  the  other  end  in  the  manner  ex- 
plained in  the  preceding  paragraph.  The  cross 

section  of  a  complete  long-distance  transmitter  is  shown  in  Fig.  341. 
387.  The  subscriber's  telephone  connections.  In  the  most  recent  prac- 
tice of  the  Bell  Telephone  Company  the  local  battery  at  the  subscriber's 
end  is  done  away  with  altogether  and  the  primary  current  is  furnished 
by  a  24-volt  battery  at  the  central  station.  Fig.  342  shows  the  essential 
elements  of  such  a  system.  When  the  subscriber  wishes  to  call  up  cen 
tral,  he  has  only  to  lift  the  receiver  from  the  hook.  This  closes  the  line 
circuit  at  t,  and  the  direct  current  which  at  once  begins  to  flow  from  the 


FIG.  341.   Cross  section  of 

a  long-distance  telephone 

transmitter 


Subscriber 


Line 


Central 


Ringing 

Calling 


FIG.  342.   The  modern  telephone  circuit  (central-station  system) 

battery  B  through  the  electromagnet  g  closes  the  circuit  of  B  through 
the  glow  lamp  /  and  the  contact  point  r.  This  lights  up  the  lamp  / 
which  is  upon  the  switchboard  in  front  of  the  operator.  Upon  seeing 
this  signal  the  latter  inserts  the  answering  plug  P  into  the  subscribers' 
"  jack  "  J  and  connects  her  own  receiver  R'  into  the  line  by  pressing 
the  listening  key  k.  The  operation  of  inserting  the  plug  P  extinguishes 
the  lamp  I  by  disconnecting  the  contact  points  o.  The  battery  B'  is, 
however,  now  upon  the  line  (B  and  B'  are,  in  fact,  one  and  the  same 
battery,  shown  here  separate  only  for  the  sake  of  simplifying  the 


INDUCTION  COIL  AND  TRANSFORMER         313 

diagram).  As,  now,  the  subscriber  talks  into  the  transmitter  T,  the 
strength  of  the  direct  current  from  the  battery  B'  through  the  primary p 
is  varied  by  the  varying  pressure  of  the  diaphragm  E  upon  the  granular 
carbon  c,  and  these  variations  induce  in  the  secondary  s  the  talking 
currents  which  pass  over  the  line  to  the  receiver  R'  of  the  operator. 
Although  with  this  arrangement  the  primary  and  secondary  currents 
pass  simultaneously  over  the  same  line,  speech  is  found  to  be  trans- 
mitted quite  as  distinctly  as  when  the  two  circuits  are  entirely  separate, 
as  is  the  case  with  the  arrangement  of  Fig.  340.  When  the  operator 
finds  what  number  the  subscriber  wishes,  she  inserts  the  calling  plug  P' 
into  the  proper  line  and  presses  the  ringing  key  k'.  This  cuts  out  the 
first  subscriber,  while  the  ringing  is  going  on,  by  opening  the  contact 
points  o'  and  closing  the  points  o".  When  the  person  called  answers, 
the  ringing  key  k'  is  released,  and  the  two  subscribers  are  thus  con- 
nected and  the  magneto  M,  which  actually  runs  all  the  time,  is  discon- 
nected from  both  lines.  Also  the  operator  releases  her  key  k  and  thus 
cuts  out  her  receiver  while  the  conversation  is  going  on.  When  one  of 
the  subscribers  "  hangs  up,"  another  lamp  like  /  is  lighted  by  a  mechanism 
not  shown  here  and  the  operator  then  pulls  out  both  plugs  P  and  P'. 

The  bell  b  rings  when  an  alternating  P.D.  is  thrown  upon  the  line, 
because,  although  the  circuit  is  broken  at  t,  an  alternating  current  will 
surge  into  and  out  of  the  condenser  C  and  thus  pull  the  armature  first 
toward  m  and  then  toward  n.  The  bell  could,  of  course,  not  be  rung  by 
a  direct  current. 

QUESTIONS  AND  PROBLEMS 
\ 

y  1.  Does  the  spark  of  an  induction  coil  occur  at  make  or  at  break? 

Why? 

3)  2.  Explain  why  an  induction  coil  is  able  to  produce  such  an  enormous 
E.M.F.  (iDraw  a  diagram  to  illustrate  the  method  of  operation  of  the  coil  J  / 
'^3.  Why  could  not  an  armature  core  be  made  of  coaxial  cylinders  of 
iron  running  the  full  length  of  the  armature,  instead  of  flat  disks,  as 
shown  in  Fig.  329  ? 

:X  4.  What  relation  must  exist  between  the  number  of  turns  on  the 
primary  and  secondary  of  a  transformer  which  feeds  110-volt  lamps 
from  a  main  line  whose  conductors  are  at  1000  volts  P.D.  ? 

5.  The  same  amount  of  power  is  to  be  transmitted  over  two  lines 
from  a  power  plant  to  a  distant  city.  If  the  heat  losses  in  the  two  lines 
are  to  be  the  same,  what  must  be  the  ratio  of  the  cross  sections  of  the 
two  lines  if  one  current  is  transmitted  at  100  volts  and  the  other  at 
10,000  volts  ? 


CHAPTER 

NATURE  AND  TRANSMISSION  OF  SOUND 
SPEED  AND  NATURE  OF  SOUND 

388.  Sources  of  sound.    If  a  sounding  tuning  fork  provided 
with  a  stylus  is  stroked  across  a  smoked-glass  plate,  it  produces 
a  wavy  line,  as  shown  in  Fig.  343 ;  if  a  light,  suspended  ball 
is  brought  into  contact  with  it,  the  latter  is  thrown  off  with 
considerable  violence.    If  we  look  about  for  the  source  of  any 
sudden  noise,  we  find  that  some  ob- 
ject has  fallen,  or  some  collision  has 

occurred,  or  some  explosion  has  taken      FlG-_ 343-  Trace  made  b7 
place ;  in  a  word,  that  some  violent 

motion  of  matter  has  been  set  up  in  some  way.  From  these 
familiar  facts  we  conclude  that  sound  arises  from  the  motions 
of  matter. 

389.  Media  of  transmission.    Air  is  ordinarily  the  medium 
through  which  sound  comes  to  our  ears,  yet  the  Indians  put 
their  ears  to  the  ground  to  hear  a  distant  noise,  and  most  boys 
know  how  loud  the  clapping  of  stones  sounds  under  water. 
If  the  base  of  the  sounding  fork  of  Fig.  343  is  held  in  a  dish 
of  water,  the  sound  will  be  markedly  transmitted  by  the  water. 
These  facts  show  that  a  gas  like  air  is  certainly  no  more 
effective  in  the  transmission   of   sound   than   a   liquid   or   a 
solid.    Let  us  next  see  whether  or  not  matter  is  necessary 
at  all  for  the  transmission  of  sound. 

*  This  chapter  should  be  accompanied  by  laboratory  experiments  on  the  speed 
of  sound  in  air,  the  vibration  rate  of  a  fork,  and  the  determination  of  wave  lengths. 
See,  for  example,  Experiments  38,  39,  and  40  of  the  authors'  manual. 

314 


SPEED  AND  NATUBE  OF  SOUND  315 

Let  an  electric  bell  be  suspended  inside  the  receiver  of  an  air  pump 
by  means  of  two  fine  springs  which  pass  through  a  rubber  stopper  in 
the  manner  shown  in  Fig.  344.  Let  the  air  be  exhausted  from  the 
receiver  by  means  of  the  pump.  The  sound  of  the 
bell  will  be  found  to  become  less  and  less  pro- 
nounced. Let  the  air  be  suddenly  readmitted.  The 
volume  of  sound  will  at  once  increase. 

Since  the  nearer  we  approach  a  vacuum, 
the  less  distinct  becomes  the  sound,  we  infer 
that  sound  cannot  be  transferred  through 
a  vacuum  and  that  therefore  the  transmis- 
sion of  sound  is  effected  only  through  the  FIG.  344.  Sound  not 
agency  of  ordinary  matter.  In  this  respect  transmitted  through 

f    ,.~          ,     '       ,  -,    r    ,  ,         f.  -,  a  vacuum 

sound  differs  from  heat  and  light,  which 

evidently  pass  with  perfect  readiness  through  a  vacuum, 
since  they  reach  the'  earth  from  the  sun  and  stars. 

390.  Speed  of  transmission.  The  first  attempt  to  measure 
accurately  the  speed  of  sound  was  made  in  1738,  when  a  com- 
mission of  the  French  Academy  of  Sciences  stationed  two 
parties  about  three  miles  apart  and  observed  the  interval 
between  the  flash  of  a  cannon  and  the  sound  of  the  report. 
By  taking  observations  between  the  two  stations,  first  in  one 
direction  and  then  in  the  other,  the  effect  of  the  wind  was 
eliminated.  A  second  commission  repeated  these  experiments 
in  1832,  using  a  distance  of  18.6  kilometers,  or  a  little  more 
than  11.5  miles.  The  value  found  was  331.2  meters  per  sec- 
ond at  0°  C.  The  accepted  value  is  now  331.3  meters.  The 
speed  in  water  is  about  1400  meters  per  second  and  in  iron 
5100  meters. 

The  speed  of  sound  in  air  is  found  to  increase  with  an  in- 
crease in  temperature.  The  amount  of  this  increase  is  about 
60  centimeters  per  degree  centigrade.  Hence  the  speed  at 
20°  C.  is  about  343.3  meters  per  second.  The  above  figures 
are  equivalent  to  1087  feet  per  second  at  0°C.,  or  1126  feet 
per  second  at  20°  C. 


316      NATUKE  AND  TBANSMISSION  OF  SOUND 

391.  Mechanism  of  transmission.  When  a  firecracker  or  toy 
cap  explodes,  the  powder  is  suddenly  changed  to  a  gas  the 
volume  of  which  is  enormously  greater  than  the  volume  of 
the  powder.  The  air  is  therefore  suddenly  pushed  back  in  all 
directions  from  the  center  of  the  explosion.  This  means  that 
the  air  particles  which  lie  about  this  center  are  given  violent 
outward  velocities.*  When  these  outwardly  impelled  air  parti- 
cles collide  with  other  particles,  they  give  up  their  outward 
motion  to  these  second  particles,  and  these  in  turn  pass  it  on 
to  others,  etc.  It  is  clear,  therefore,  that  the  motion  started  by 
the  explosion  must  travel  on  from  particle  to  particle  to  an 
indefinite  distance  from  the  center  of  the  explosion.  Further- 
more, it  is  also  clear  that,  although  the  motion  travels  on  to 
great  distances,  the  individual  particles  do  not  move  far  from 
their  original  positions ;  for  it  is  easy 
to  show  experimentally  that  whenever 
an  elastic  body  in  motion  collides  with 
another  similar  body  at  rest,  the  collid- 
ing body  simply  transfers  its  motion  to 
the  body  at  rest  and  comes  itself  to  rest. 

FIG.  345.   Illustrating  the 
Let  six  or  eight  equal  steel  balls  be  hung     propagation  of  sound  f  rom 

from  cords  in  the  manner  shown  in  Fig.  345.  particle  to  particle 

First,  let  all  of  the  balls  but  two  adjacent 

ones  be  held  to  one  side,  and  let  one  of  these  two  be  raised  and  allowed 
to  fall  against  the  other.  The  first  ball  will  be  found  to  lose  its  motion 
in  the  collision,  and  the  second  will  be  found  to  rise  to  practically  the 
same  height  as  that  from  which  the  first  fell.  Next,  let  all  of  the  balls  be 
placed  in  line  and  the  end  one  raised  and  allowed  to  fall  as  before.  The 
motion  will  be  transmitted  from  ball  to  ball,  each  giving  up  the  whole 
of  its  motion  practically  as  soon  as  it  receives  it,  and  the  last  ball  will 
move  on  alone  with  the  velocity  which  the  first  ball  originally  had. 

*  These  outward  velocities  are  simply  superposed  upon  the  velocities  of  agita- 
tion which  the  molecules  already  have  on  account  of  their  temperature.  For  our 
present  purpose  we  may  ignore  entirely  the  existence  of  these  latter  velocities  and 
treat  the  particles  as  though  they  were  at  rest,  save  for  the  velocities  imparted 
by  the  explosion. 


SPEED  AND  NATURE  OF  SOUND  317 

The  preceding  experiment  furnishes  a  very  nice  mechanical 
illustration  of  the  manner  in  which  the  air  particles  which 
receive  motions  from  an  exploding  firecracker  or  a  vibrating 
tuning  fork  transmit  these  motions  in  all  directions  to  neigh- 
boring layers  of  air,  these  in  turn  to  the  next  adjoining  layers, 
etc.,  until  the  motion  has  traveled  to  very  great  distances, 
although  the  individual  particles  themselves  move  only  very 
minute  distances.  When  a  motion  of  this  sort,  transmitted 
by  air  particles,  reaches  the  drum  of  the  ear,  it  produces  the 
sensation  which  we  call  sound. 

392.  A  train  of  waves  ;  wave  length.  In  the  preceding  para- 
graphs we  have  confined  attention  to  a  single  pulse  traveling 
out  from  a  center  of  explosion.  Let  us  next  consider  the  sort 
of  disturbance  which  is  set  up  in  the  air  by  a  continuously 
vibrating  body,  like  the  prong  of  Fig.  346.  Each  time  that 
this  prong  moves  to  the  right 
it  sends  out  a  pulse  which 

travels   through  the    air   at 

,     °1A_     ,  Wavelength 

the  rate   of    1100   feet   per      m 

second,  in  exactly  the  man- 

T          -i      i    •      ,1  i       FIG.  346.   Vibrating  reed  sending  out  a 

ner  described  in  the  preced-  train  of  equ^istant  pulses* 

ing    paragraphs.    Hence,    if 

the  prong  is  vibrating  uniformly,  we  shall  have  a  continuous 
succession  of  pulses  following  each  other  through  the  air  at 
exactly  equal  intervals.  Suppose,  for  example,  that  the  prong 
makes  110  complete  vibrations  per  second.  Then  at  the  end 
of  one  second  the  first  pulse  sent  out  will  have  reached 
a  distance  of  1100  feet.  Between  this  point  and  the  prong 
there  will  be  110  pulses  distributed  at  equal  intervals;  that 
is,  each  two  adjacent  pulses  will  be  just  10  feet  apart.  If 
the  prong  made  220  vibrations  per  second,  the  distance  be- 
tween adjacent  pulses  would  be  5  feet,  etc.  The  distance 
between  two  adjacent  pulses  in  such  a  train  of  waves  is  called  a 
wave  length. 


318      NATURE  AND  TRANSMISSION  OF  SOUND 

393.  Relation  between  velocity,  wave  length,  and  number 
of  vibrations  per  second.    If  n  represents  the  number  of  vibra- 
tions per  second  of  a  source  of  sound,  I  the  wave  length,  and  v 
the  velocity  with  which  the  sound  travels  through  the  medium, 
it  is  evident  from  the  example  of  the  preceding  paragraph  that 
the  following  relation  exists  between  these  three  quantities : 

I  =  v/n,  or  v  =  nl\  (1) 

that  is,  wave  length  is  equal  to  velocity  divided  by  the  number  of 
vibrations  per  second,  or  velocity  is  equal  to  the  number  of  vibra- 
tions per  second  times  the  wave  length. 

394.  Condensations  and  rarefactions.    Thus  far,  for  the  sake 
of  simplicity,  we  have  considered  a  train  of  waves  as  a  series 
of  thin,  detached  pulses  separated  by  equal  intervals  of  air  at 
rest.    In  point  of  fact,  however,  the  air  in  front  of  the  prong 
B  (Fig.  346)  is  being  pushed  forward  not  at  one  particular 


FIG.  347.    Illustrating  motions  of  air  particles  in  one  complete  sound  wave 
consisting  of  a  condensation  and  a  rarefaction 

instant  only,  but  during  all  the  time  that  the  prong  is  moving 
from  A  to  (7,  that  is,  through  the  time  of  one  half  vibration  of 
the  fork ;  and  during  all  this  time  this  forward  motion  is  being 
transmitted  to  layers  of  air  which  are  farther  and  farther  away 
from  the  prong,  so  that  when  the  latter  reaches  (7,  all  the  air 
between  C  and  some  point  c  (Fig.  347)  one-half  wave  length 
away  is  crowding  forward,  and  is  therefore  in  a  state  of  com- 
pression or  condensation.  Again,  as  the  prong  moves  back  from 
C  to  -4,  since  it  tends  to  leave  a  vacuum  behind  it,  the  adja- 
cent layer  of  air  rushes  in  to  fill  up  this  space,  the  layer  next 
adjoining  follows,  etc.,  so  that  when  the  prong  reaches  A,  all 
the  air  between  A  and  c  (Fig.  347)  is  moving  backward  and 


SPEED  AND  NATURE  OF  SOUND 


319 


is  therefore  in  a  state  of  diminished  density  or  rarefaction. 

During  this  time  the  preceding  forward  motion  has  advanced 

one-half  wave  length  to  the  right,  so  that  it  now  occupies  the 

region  between  c  and  a  (Fig.  347).    Hence  at  the  end  of  one 

complete  vibration  of  the  prong  we  may  divide  the  air  between 

it  and  a  point  one  wave 

length  away  into  two 

portions,  one  a  region 

of  condensation  ac,  and 

the  other  a  region  of 

rarefaction^.  Thear-         abcdefghij 

rows  in  Fig.  347  rep-  FIG.  348.   Illustration  of  sound  waves 

resent  the  direction  and  relative  magnitudes  of  the  motions 

of  the  air  particles  in  various  portions  of  a  complete  wave. 

At  the  end  of  n  vibrations  the  first  disturbance  will  have 
reached  a  distance  n  wave  lengths  from  the  fork,  and  each  wave 
between  this  point  and  the  fork  will  consist  of  a  condensation 
and  a  rarefaction,  so  that  sound  waves  may  be  said  to  consist 
of  a  series  of  condensations  and  rarefactions  following  one 
another  through  the  air  in  the  manner  shown  in  Fig.  348. 

Wave  length  may  now  be  more  accurately  defined  as  the 
distance  between  two  successive  points  of  maximum  condensation 
(b  and/,  Fig.  348)  or  of  maximum  rarefaction  (d  and  h~). 

395.  Water-wave  analogy.  Condensations  and  rarefactions 
of  sound  waves  are  exactly  analogous  to  the  familiar  crests  and 
troughs  of  water  waves.' 
Thus  the  wave  length  of 
such  a  series  of  waves  as 
that  shown  in  Fig.  349 
is  defined  as  the  distance 
bf  between  two  crests,  or  the  distance  dh,  or  ae,  or  eg,  or  mn, 
between  any  two  points  which  are  in  the  same  condition  or  phase 
of  disturbance.  The  crests,  that  is,  the  shaded  portions,  which 
are  above  the  natural  level  of  the  water,  correspond  exactly 


d  fi 

FIG.  349.   Illustrating  wave  length  of 
water  waves 


320      NATURE  AND  TRANSMISSION  OF  SOUND 

to  the  condensations  of  sound  waves,  that  is,  to  the  portions  of 
air  which  are  above  the  natural  density.  The  troughs,  that  is, 
the  dotted  portions,  correspond  to  the  rarefactions  of  sound 
waves,  that  is,  to  the  portions  of  air  which  are  below  the  nat- 
ural density.  Nevertheless,  the  analogy  breaks  down  at  one 
point ;  for  in  water  waves  the  motion  of  the  particles  is  trans- 
verse to  the  direction  of  propagation,  while  in  sound  waves,  as 
shown  in  §  394,  the  particles  move  back  and  forth  in  the  line 
of  propagation  of  the  wave.  Water  waves  are  therefore  called 
transverse  waves,  while  sound  waves  in  air  are  longitudinal  waves. 

396.  Distinction  between  musical   sounds   and  noises.    Let 

a  current  of  air  from  a  |pinch  nozzle  be  directed  against  a  row  of 
forty-eight  equidistant  ^-inch  holes  in  a  metal 
or  cardboard  disk,  mounted  as  in  Fig.  350  and 
set  into  rotation  either  by  hand  or  by  an  elec- 
tric motor.  A  very  distinct  musical  tone  will 
be  produced.  Then  let  the  jet  of  air  be  directed 
against  a  second  row  of  forty-eight  holes,  which 
differs  from  the  first  only  in  that  the  holes  are 
irregularly  instead  of  regularly  spaced  about 
the  circumference  of  the  disk.'  The  musical 
character  of  the  tone  will  altogether  disappear. 

The  experiment  furnishes  a  very 
striking  illustration  of  the  difference  be- 
tween a  musical  sound  and  a  noise. 
Only  those  sounds  possess  a  musical  qual-  FIG.  350.  Regularity  of 

ity  which  come  from  sources  capable  of    Pulses  the  condition  for 

V  , .  *  a  musical  tone 

sending  out  pulses,  or  waves,  at  absolutely 

regular  intervals.  Therefore  it  is  only  sounds  possessing  a 
musical  quality  which  may  be  said  to  have  wave  lengths. 

397.  Pitch.    While  the  apparatus  of  the  preceding  experiment  is 
rotating  at  constant  speed,  let  a  current  of  air  be  directed  first  against 
the  outside  row  of  regularly  spaced  holes  and  then  suddenly  turned 
against  the  inside  row,  which  is  also  regularly  spaced  but  which  contains 
a  smaller  number  of  holes.    The  note  produced  in  the  second  case  will 


SPEED  AND  NATURE  OF  SOUND  321 

be  found  to  have  a  markedly  lower  pitch  than  the  other  one.  Again, 
let  the  jet  of  air  be  directed  against  one  particular  row,  and  let  the  speed 
of  rotation  be  changed  from  very  slow  to  very  fast.  The  note  produced 
will  gradually  rise  in  pitch. 

We  conclude,  therefore,  that  the  pitch  of  a  musical  note  de- 
pends simply  upon  the  number  of  pulses  which  strike  the  ear  per 
second.  If  the  sound  comes  from  a  vibrating  body,  the  pitch  of 
the  note  depends  upon  the  rate  of  vibration  of  the  body. 

398.  Doppler's   principle.  *When   a   rapidly   moving   express   train 
rushes  past  an  observer,  he  notices  a  very  distinct  change  in  the  pitch 
of  the  bell  as  the  engine  passes  him,  the  pitch  being  higher  as  the  engine 
approaches  than  as  it  recedes.   The  explanation  is  as  follows  :    The  bell, 
of  course,  sends  out  pulses  at  exactly  equal  intervals  of  time.    As  the 
train  is  approaching,  however,  the  pulses  reach  the  ear  at  shorter  inter- 
vals than  the  intervals  between  emissions,  since  the  train  comes  toward 
the  observer  between  two  successive  emissions.    But  as  the  train  recedes, 
the  interval  between  the  receipt  of  pulses  by  the  ear  is  longer  than  the 
interval  between  emissions,  since  the  train  is  moving  away  from  the  ear 
during  the  interval  between  emissions.    Hence  the  pitch  of  the  bell  is 
higher  during  the  approach  of  the  train  than  during  its  recession.    This 
phenomenon  of  the  change  in  pitch  of  a  note  proceeding  from  an  ap- 
proaching or  receding  body  is  known  as  Doppler's  principle. 

399.  Loudness.    The  loudness  or  intensity  of  a  sound  de- 
pends upon  the  rate  at  which  energy  is  communicated  by  it 
to  the  tympanum  of  the  ear.    Loudness  is  therefore  determined 
by  the  distance  of  the  source  and  the  amplitude  of  its  vibration. 

If  a  given  sound  pulse  is  free  to  spread  equally  in  all  direc- 
tions, at  a  distance  of  100  feet  from  the  source  the  same  energy 
must  be  distributed  over  a  sphere  of  four  times  as  large  an 
area  as  at  a  distance  of  50  feet.  Hence  under  these  ideal  coiir 
ditions  the  intensity  of  a  sound  varies  inversely  as  the  square 
of  the  distance  from  the  source.  But  when  sound  is  confined 
within  a  tube  so  that  the  energy  is  continually  communicated 
from  one  layer  to  another  of  equal  area,  it  will  travel  to  great 
distances  with  little  loss  of  intensity.  This  explains  the  effi- 
ciency of  speaking  tubes  and  megaphones. 


322      NATURE  AND  TRANSMISSION  OF  SOUND 


QUESTIONS  AND   PROBLEMS 

1.  A  thunderclap  was  heard  5J  sec.  after  the  accompanying  light- 
ning flash  was  seen.    How  far  away  did  the  flash  occur  ? 

2.  A  bullet  fired  from  a  rifle  with  a  speed  of  1200  ft.  per  second  is 
heard  to  strike  the  target  6  sec.  afterwards.    What  is  the  distance  to 
the  target,  the  temperature  of  the  air  being  20°  C.  ? 

3.  A  church  bell  is  ringing  at  a  distance  of  ^  ini.  from  one  man 
and  \  mi.  from  another.    How  much  louder  would  it  appear  to  the 
second  man  than  to  the  first,  if  no  reflections  of  the  sound  took  place? 

4.  Explain  the  principle  of  the  ear  trumpet. 

5.  The  vibration  rate  of  a  fork  is  256.    Find  the  wave  length  of  the 
note  given  out  by  it  at  20° C. 

6.  A  stone  is  dropped  into  a  well  200m.  deep.    At  20°  C.  how  much 
time  will  elapse  before  the  sound  of  the  splash  is  heard  at  the  top  ? 

7.  As  a  circular  saw  cuts  into  a  block  of  wood  the  pitch  of  the  note 
given  out  falls  rapidly.    Why  ? 

8.  Since  the  music  of  an  orchestra  reaches  a  distant  hearer  without 
confusion  of  the  parts,  what  may  be  inferred  as  to  the  relative  velocities 
of  the  notes  of  different  pitch  ? 


REFLECTION,  REENFORCEMENT,  AND  INTERFERENCE 

400.  Echo.  That  a  sound  wave  in  hitting  a  wall  suffers 
reflection  is  shown  by  the  familiar  phenomenon  of  echo.  The 
roll  of  thunder  is  due  to  successive  reflections  of  the  original 
sound  from  clouds  and  other  surfaces  which  are  at  different 
distances  from  the  observer. 

In  ordinary  rooms  the  walls  are  so  close  that  the  reflected 
waves  return  before  the  effect  of  the  original  sound  on  the 
ear  has  died  out.  Consequently  the  echo  blends  with  and 
strengthens  the  original  sound  instead  of  interfering  with  it. 
This  is  why,  in  general,  a  speaker  may  be  heard  so  much 
better  indoors  than  in  the  open  ah-.  Since  the  ear  cannot 
appreciate  successive  sounds  as  distinct  if  they  come  at  inter- 
vals shorter  than  a  tenth  of  a  second,  it  will  be  seen  from  the 
fact  that  sounds  travels  about  113  feet  in  a  tenth  of  a  second 
that  a  wall  which  is  nearer  than  about  50  feet  cannot  possibly 


REFLECTION  AND  REENFORCEMENT 


323 


produce  a  perceptible  echo.  In  rooms  which  are  large  enough 
to  give  rise  to  troublesome  echoes  it  is  customary  to  hang 
draperies  of  some  sort,  so  as  to  break  up  the  sound  waves 
and  prevent  regular  reflection. 


1 


FIG.  351.    Sound  foci 


401.  Sound  foci.    Let  a  watch  be  hung  at  the  focus  of  a  large  con- 
cave mirror.   On  account  of  the  reflection  from  the  surface  of  the  mirror 
a  fairly  well-defined  beam  of  sound  will 

be  thrown  out  in  front  of  the  mirror, 

so  that  if  both  watch  and  mirror  are 

hung  on  a  single  support  and  the  whole 

turned  in  different  directions  toward  a 

number  of  observers,  the  ticking  will 

be  distinctly  heard  by  those  directly  in  front  of  the  mirror,  but  not  by 

those  at  one  side.    If  a  second  mirror  is  held  in  the  path  of  this  beam, 

as  in  Fig.  351,  the  sound  may  be  again  brought  to  a  focus,  so  that  if  the 

ear  is  placed  in  the  focus  of  this  second  mirror,  or  better  still,  if  a  small 

funnel  which  is  connected  with  the  ear  by  a  rubber  tube  is  held  in  this 

focus,  the  ticking  of  the  watch  may  sometimes  be  heard  hundreds  of  feet 

away.   A  whispering  gallery  is  a  room  so  arranged 

as  to  contain  such  sound  foci.    Any  two  opposite 

points  a  few  feet  from  the  walls  of  a  dome,  like 

that  of  St.  Peter's  at  Rome  or  St.  Paul's  at  London, 

are  sufficiently  near  to  such  sound  foci  to  make 

very  low  whispers  on  one  side  distinctly  audible 

at  the  other,  although  at  intermediate  points  no 

sound  can  be  heard. 

402.  Resonance.    Resonance  is  the  reen- 
forcement  or  intensification  of  sound  because 
of  the  union  of  direct  and  reflected  waves. 

Thus  let  one  prong  of  a  vibrating  tuning  fork, 
which  makes,  for  example,  512   vibrations  per 
second,  be  held  over  the  mouth  of  a  tube  an  inch     -plG   352    Illustrating 
or  so  in  diameter,  arranged  as  in  Fig.  352,  so  that  resonance 

as  the  vessel  A  is  raised  or  lowered,  the  height  of 

the  water  in  the  tube  may  be  adjusted  at  will.  It  will  be  found  that  as 
the  position  of  the  water  is  slowly  lowered  from  the  top  of  the  tube  a 
very  marked  reenforcement  of  the  sound  will  occur  at  a  certain  point. 

t 


^r  A 


324      NATUBE  AND  TRANSMISSION  OF  SOUND 


Let  other  forks  of  different  pitch  be  tried  in  the  same  way.  It  will 
be  found  that  the  lower  the  pitch  of  the  fork,  the  lower  must  be  the 
water  in  the  tube  in  order  to  get  the  best  reenforcement.  This  means 
that  the  longer  the  wave  length  of  the  note  which  the  fork  produces, 
the  longer  must  be  the  air  column  in  order  to  obtain  resonance. 

We  conclude,  therefore,  that  a  fixed  relation  exists  between 
the  ivave  length  of  a  note  and  the  length  of  the  air  column  which 
will  reenforce  it. 

403.  Best  resonant  length  of  a  closed  pipe  is  one-fourth  wave 
length.  If  we  calculate  the  wave  length  of  the  note  of  the 
fork  by  dividing  the  speed  of  sound  by  the  vibration  rate  of 
the  fork,  we  shall  find  that,  in  every  case,  the 
length  of  air  column  which  gives  the  best  response 
is  approximately  one-fourth  wave  length.  The 
reason  for  this  is  evident  when  we  consider 
that  the  length  must  be  such  as  to  enable  the 
reflected  wave  to  return  to  the  mouth  just  in 
time  to  unite  with  the  direct  wave  which  is  at 
that  instant  being  sent  off  by  the  prong.  Thus 
when  the  prong  is  first  starting  down  from  the 
position  A  (see  Fig.  358),  it  starts  the  begin- 
ning of  a  condensation  down  the  tube.  If  this 
motion  is  to  return  to  the  mouth  just  in  time 
to  unite  with  the  direct  wave  sent  off  by  the 
prong,  it  must  get  back  at  the  instant  that  the 
prong  is  first  starting  up  from  the  position  C.  In 


FIG.  353.    Reso- 
nant length  of  a 
closed  pipe  is  ^ 
wave  length 


other  words,  the  pulse  must  go  down  the  tube  and  back  again 
while  the  prong  is  making  a  half  vibration.  This  means -that 
the  path  down  and  back  must  be  a  half  wave  length,  and  hence 
that  the  length  of  the  tube  must  be  a  fourth  of  a  wave  length. 
From  the  above  analysis  it  will  appear  that  there  should 
also  be  resonance  if  the  reflected  wave  does  not  return  to  the 
mouth  until  the  fork  is  starting  back  its  second  time  from  C1 
that  is,  at  the  end  of  one  and  a  half  vibrations  instead  of  a 


BEFLECTION  AKD  KEENFORCEMENT  325 

half  vibration.  The  distance  from  the  fork  to  the  water  and 
back  would  then  be  one  and  a  half  wave  lengths ;  that  is,  the 
water  surface  would  be  a  half  wave  length  farther  down  the 
tube  than  at  first.  The  tube  length  would  therefore  now  be 
three  fourths  of  a  wave  length. 

Let  the  experiment  be  tried.  A  similar  response  will  indeed  be 
found,  as  predicted,  a  half  wave  length  farther  dow,n  the  tube.  This 
response  will  be  somewhat  weaker  than  before,  as  the  wave  has  lost 
some  of  its  energy  in  traveling  a  long  distance  through  the  tube.  It 
may  be  shown  in  a'  similar  way  that  there  will  be  resonance  where  the 
tube  length  is  |,  5,  or  indeed  any  odd  number  of  quarter  wave  lengths. 

404.  Best  resonant  length  of  an  open  pipe  is  one-half  wave 

length.    Let  the  same  tuning  fork  which  was  used  in  §  403  be  held  in 
front  of  an  open  pipe  (8  or  10  inches 
long)  the  length  of  which  is  made  ad- 
justable by  slipping  back  and  forth  over 

it  a  tightly  fitting  roll  of  writing  paper 

/T7-     o~^     -n.     mv    f        1*1    4.V  TIG.  354.   Resonant  length  of  an 

(Fig.  3o4).   It  will  be  found  that  for  one         open  pipe  ig  ,  waye  length 

particular  length  this  open  pipe  will  re- 
spond quite  as  loudly  as  did  the  closed  pipe,  but  the  responding  length 
loill  be  found  to  be  just  twice  as  great  as  before.    Other  resonant  lengths 
can  be  found  when  the  tube  is  made  2,  3,  etc.,  times  as  long. 

We  learn,  then,  that  the  shortest  resonant  length  of  an  open 
pipe  is  one-half  wave  length,  and  that  there,  is  resonance  at  any 
multiple  of  a  half  wave  length. 

The  fact  that  the  shortest  resonant  length  of  the  open  pipe 
is  just  twice  that  of  the  closed  one  is  the  experimental  proof 
that  a  condensation,  upon  reaching  the  open  end  of  a  pipe,  is 
reflected  as  a  rarefaction.  This  means  that  when  the  lower 
end  of  the  tube  of  Fig.  353  is  open,  a  condensation  upon 
reaching  it  suddenly  expands.  In  consequence  of  this  expan- 
sion the  new  pulse  which  begins  at  this  instant  to  travel  back 
through  the  tube  is  one  in  which  the  particles  are  moving 
down  instead  of  up ;  that  is,  the  particles  are  moving  in  a 
direction  opposite  to  that  in  which  the  wave  is  traveling. 
This  is  always  the  case  in  a  rarefaction  (see  Fig.  347).  In 


326      NATURE  AND  TRANSMISSION  OF  SOUND 

order  then  to  unite  with  the  motion  of  the  prong  this  down- 
ward motion  of  the  particles  must  get  back  to  the  mouth 
when  the  prong  is  just  starting  down  from  A  the  second  time ; 
that  is,  after  one  complete  vibration  of  the  prong.  This  shows 
why  the  pipe  length  is  one-half  wave  length. 

405.  Resonators.    If  the  vibrating  fork  at  the  mouth  of  the 
tubes  in  the  preceding  experiments  is  replaced  by  a  train  of 
waves  coming  from  a  distant  source,  precisely  the  same  analysis 
leads  to  the  conclusion  that  the  waves  reflected  from  the  bottom 
of  the  tube  will  reenforce  the  oncoming  waves  when  the  length 
of  the  tube  is  any  odd  number  of  quarter  wave  lengths  in  the 
case  of  a  closed  pipe,  or  any  number  of  half  wave  lengths  in  the 
case  of  an  open  pipe.    It  is  clear,  therefore,  that  every  air  cham- 
ber will  act  as  a  resonator  for  trains  of  waves  of  a  certain  wave 
length.    This  is  why  a  conch  shell  held  to  the  ear  is  always 
heard  to  hum  with  a  particular  note.   Feeble  waves  which  pro- 
duce no  impression  upon  the  unaided  ear  gain  sufficient  strength 
when  reenforced  by  the  shell  to  become  audible.   When  the  air 
chamber  is  of  irregular  form  it  is  not  usually  possible  to  calcu- 
late to  just  what  wave  length  it  will  respond,  but  it  is  always 
easy  to  determine  experimentally  what  particular  wave  length 
it  is  capable  of  reenforcing.   The  resonators  on  which  tuning 
forks  are  mounted  are  air  chambers  which  are  of  just  the  right 
dimensions  to  respond  to  the  note  given  out  by  the  fork. 

406.  Forced  vibrations ;  sounding  boards.    Let  a  tuning  fork 

be  struck  and  held  in  the  hand.  The  sound  will  be  entirely  inaudible 
except  to  those  quite  near.  Let  the  base  of  the  sounding  fork  be  pressed 
firmly  against  the  table.  The  sound  will  be  found  to  be  enormously 
intensified.  Let  another  fork  be  held  against  the  same  table.  Its  sound 
will  also  be  reenforced.  In  this  case,  then,  the  table  intensifies  the  sound 
of  any  fork  which  is  placed  against  it,  while  an  air  column  of  a  certain 
size  could  intensify  only  a  single  note. 

The  cause  of  the  response  in  the  two  cases  is  wholly  differ- 
ent.  In  the  last  case  the  vibrations  of  the  fork  are  transmitted 


REFLECTION  AND  KEENFOKCEMENT  327 

through  its  base  to  the  table  top  and  force  the  latter  to  vibrate 
in  its  own  period.  The  vibrating  table  top,  on  account  of  its 
large  surface,  sets  a  comparatively  large  mass  of  air  into  motion 
and  therefore  sends  a  wave  of  great  intensity  to  the  ear ;  while 
the  fork  alone,  with  its  narrow  prongs,  was  not  able  to  impart 
much  energy  to  the  air.  Vibrations  like  those  of  the  table  top 
are  called  forced  because  they  can  be  produced  with  any  fork, 
no  matter  what  its  period.  Sounding  boards  in  pianos  and 
other  stringed  instruments  act  precisely  as  does  the  table  top 
in  this  experiment ;  that  is,  they  are  set  into  forced  vibrations 
by  any  note  of  the  instrument  and  reenforce  it  accordingly. 

407.  Beats.  Since  two  sound  waves  are  able  to  unite  so  as 
to  reenforce  each  other,  it  ought  also  to  be  possible  to  make 
them  unite  so  as  to  interfere  with  or  destroy  each  other.  -  In 
other  words,  under  the  proper  conditions  the  union  of  two 
sounds  ought  to  produce  silence. 

Let  two  mounted  tuning  forks  of  the  same  pitch  be  set  side  by  side, 
as  in  Fig.  355.  Let  the  two  forks  be  struck  in  quick  succession  with  a 
soft  mallet,  for  example,  a  rubber  stopper  on  the  end  of  a  rod.  The  two 
notes  will  blend  and  produce  a  smooth,  even  tone.  Then  let  a  piece  of  wax 
or  a  small  coin  be  stuck  to  a  prong 
of  one  of  the  forks.  This  dimin- 
ishes slightly  the  number  of  vibra- 
tions which  this  fork  makes  per 
second,  since  it  increases  its  mass. 
Again,  let  the  two  forks  be  sounded  FlG"  355'  Arrangement  of  forks 

^,  for  beats 

together.    The  former  smooth  tone 

will  be  replaced  by  a  throbbing  or  pulsating  one.  This  is  due  to  the 
alternate  destruction  and  reinforcement  of  the  sounds  produced  by 
the  two  forks.  This  pulsation  is  called  the  phenomenon  of  beats. 

The  mechanism  of  the  alternate  destruction  and  reenforce- 
ment  may  be  understood  from  the  following.  Suppose  that  one 
fork  makes  256  vibrations  per  second  (see  the  dotted  line  AC 
in  Fig.  356),  while  the  other  makes  255  (see  the  heavy  line 
AC).  If  at  the  beginning  of  a  given  second  the  two  forks 


328      NATURE  AND  TRANSMISSION  OF  SOUND 

are  swinging  together  so  that  they  simultaneously  send  out 
condensations  to  the  observer,  these  condensations  will  of 
course  unite  so  as  to  produce  a  double  effect  upon  the  ear 
(see  A',  Fig.  356).  Since  now  one  fork  gains  one  complete 
vibration  per  second  over  the  other,  at  the  end  of  the  second 
considered  the  two  forks  A  B  c 

will    again    be    vibrating 
together,  that  is,  senduig      , 
out    condensations    which 
add  their  effects  as  before 

(see  C'}.   In  the  middle  of 

FIG.  356.    Graphical  illustration  of  beats 

this  second,  however,  the 

two  forks  are  vibrating  in  opposite  directions  (see  B)  ;  that 
is,  one  is  sending  out  rarefactions  while  the  other  sends  out 
condensations.  At  the  ear  of  the  observer  the  union  of  the 
rarefaction  (backward  motion  of  the  air  particles)  produced 
by  one  fork  with  the  condensation  (forward  motion)  pro- 
duced by  the  other  results  in  no  motion  at  all,  provided  the 
two  motions  have  the  same  energy ;  that  is,  in  the  middle  of 
the  second  the  two  sounds  have  united  to  produce  silence  (see  B1). 
It  will  be  seen  from  the  above  that  the  number  of  beats  per  second 
is  equal  to  the  difference  in  the  vibration  numbers  of  the  two  forks. 

To  test  this  conclusion,  let  more  wax  or  a  heavier  coin  be  added  to 
the  weighted  prong ;  the  number  of  beats  per  second  will  be  increased. 
Diminishing  the  weight  will  reduce  the  number  of  beats  per  second. 

408.  Interference  of  sound  waves  by  reflection.  Let  a  thin 
cork  about  an  inch  in  diameter  be  attached  to  one  end  of  a  brass  or 


FIG.  357.    Interference  of  advancing  and  retreating  trains  of  sound  waves 

glass  rod  from  one  to  two  meters  long.  Let  this  rod  be  clamped  firmly 
in  the  middle,  as  in  Fig.  357.  Let  a  piece  of  glass  tubing  a  meter  or 
more  long  and  from  an  inch  to  an  inch  and  a  half  in  diameter  be  slipped 


REFLECTION  AND  REENFORCEMENT  329 

over  the  cork,  as  shown.  Let  the  end  of  the  rod  be  stroked  longitudi- 
nally with  a  well-resined  cloth.  (A  wet  cloth  will  answer  better  if  the 
rod  is  of  glass.)  A  loud  shrill  note  will  be  produced. 

This  note  is  due  to  the  fact  that  the  slipping  of  the  resined  cloth  over 
the  surface  of  the  rod  sets  the  latter  into  longitudinal  vibrations,  so  that  its 
ends  impart  alternate  condensations  and  rarefactions  to  the  layers  of  air 
in  contact  with  them.  As  soon  as  this  note  is  started  the  cork  dust  inside 
the  tube  will  be  seen  to  be  intensely  agitated.  If  the  effect  is  not  marked 
at  first,  a  slight  slipping  of  the  glass  tube  forward  or  back  will  bring  it  out. 
Upon  examination  it  will  be  seen  that  the  agitation  of  the  cork  dust  is  not 
uniform,  but  at  regular  intervals  throughout  the  tube  there  will  be  regions 
of  complete  rest,  nv  n2,  n3,  etc.,  separated  by  regions  of  intense  motion. 

The  points  of  rest  correspond  to  the  positions  in  which  the 
reflected  train  of  sound  waves  returning  from  the  end  of  the 
tube  neutralizes  the  effect  of  the  advancing  train  passing  down 
the  tube  from  the  vibrating  rod.  The  points  of  rest  are  called 
nodes,  the  intermediate  a^  ^  ^  ^ 

portions   loops  or   anti-     =ftR      I  II  II 

nodes.    The  distance  be-  ^ 

tween  these  nodes  is  one    Fi-' 358'  Distance  bet1ween1  nodes  is  one  half 

wave  length 
half  wave  length,  for  at 

the  instant  that  the  first  wave  front  ax  (Fig.  358)  reaches  the 
end  of  the  tube  it  is  reflected  and  starts  back  toward  R.  Since 
at  this  instant  the  second  wave  front  «2  is  just  one  wave  length 
to  the  left  of  a^  the  two  wave  fronts  must  meet  each  other  at 
a  point  nv  just  one-half  wave  length  from  the  end  of  the  tube. 
The  exactly  equal  and  opposite  motions  of  the  particles  in  the 
two  wave  fronts  exactly  neutralize  each  other.  Hence  the  point 
n^  is  a  point  of  no  motion,  that  is,  a  node.  Again,  at  the  in- 
stant that  the  reflected  wave  front  al  met  the  advancing  wave 
front  &2  at  nr  the  third  wave  front  az  was  just  one  wave  length 
to  the  left  of  nr  Hence,  as  the  first  wave  front  a^  continues 
to  travel  back  toward  R  it  meets  a3  at  n2,  just  one  half  wave 
length  from  n^  and  produces  there  a  second  node.  Similarly, 
a  third  node  is  produced  at  nffl  one  half  wave  length  to  the 


330      NATURE  AND  TRANSMISSION  OF  SOUND 

left  of  wa,  etc.  Thus  the  distance  between  two  nodes  must  always 
be  just  one  half  the  wave  length  of  the  waves  in  the  train. 

In  the  preceding  discussion  it  has  been  tacitly  assumed  that  the  two 
oppositely  moving  waves  are  able  to  pass  through  each  other  without 
either  of  them  being  modified  by  the  presence  of  the  other.  That  two 
opposite  motions  are,  in  fact,  transferred  in  just  this  manner  through  a 
medium  consisting  of  elastic  particles  may  be  beautifully  shown  by  the 
following  experiment  with  the  row  of  balls  used  in  §  391. 

Let  the  ball  at  one  end  of  the  row  be  raised  a  distance  of,  say,  2  inches 
and  the  ball  at  the  other  end  raised  a  distance  of  4  inches.  Then  let 
both  balls  be  dropped  simultaneously  against  the  row.  The  two  opposite 
motions  will  pass  through  each  other  in  the  row  altogether  without 
modification,  the  larger  motion  appearing  at  the  end  opposite  to  that  at 
which  it  started,  and  the  smaller  likewise. 

Another  and  more  complete  analogy  to  the  condition  existing  within 
the  tube  of  Fig.  357  may  be  had  by  simply  vibrating  one  end  of  a  two-  or 
three-meter  rope,  as  in 
Fig.  359.  The  trains  of 
advancing  and  reflected 
waves  which  continu-  FIG.  359-  Nodes  and  loops  in  a  cord 

ously  travel  through  each  Black  line  denotes  advancing  train;  dotted  line, 
other  up  and  down  the  reflected  train 

rope  will  unite  so  as  to  form  a  series  of  nodes  and  loops.  The  nodes  at 
c  and  e  are  the  points  at  which  the  advancing  and  reflected  waves  are 
always  urging  the  cord  equally  in  opposite  directions.  The  distance 
between  them  is  one  half  the  wave  length  of  the  train  sent  down  the 
rope  by  the  hand. 

QUESTIONS  AND  PROBLEMS 

1.  Why  do  the  echoes  which  are  prominent  in  empty  halls  often 
disappear  when  the  hall  is  full  of  people  ? 

2.  A  gunner  hears  an  echo  5^  sec.  after  he  fires.   How  far  away  was 
the  reflecting  surface,  the  temperature  of  the  air  being  20° C.? 

3.  Find  the  number  of  vibrations  per  second  of  a  fork  which,  at  20°  C., 
produces  resonance  in  a  pipe  1  ft.  long  closed  at  one  end. 

4.  A  fork  making  500  vibrations  per  second  is  found  to  produce 
resonance  in  an  air  column  like  that  shown  in  Fig.  352,  first  when  the 
water  is  a  certain  distance  from  the  top,  and  again  when  it  is  34  cm. 
lower.    Find  the  velocity  of  sound. 

5.  Show  why  an  open  pipe  needs  to  be  twice  as  long  as  a  closed  pipe 
if  it  is  to  respond  to  the  same  note. 


CHAPTER  XVII 


PROPERTIES  OF  MUSICAL  SOUNDS 

MUSICAL  SCALES 
409.  Physical  basis  of  musical  intervals.   Let  a  metal  or  card 

board  disk  10  or  12  inches  in  diameter  be  provided  with  four  concentric 
rows  of  equidistant  holes,  the  successive  rows  containing  respectively 
24,  30,  36,  and  48  holes  (Fig.  360).  The  holes  should  be  about  £  inch 
in  diameter  and  the  rows  should  be  about 
|  inch  apart.  Let  the  disk  be  placed  in  the 
rotating  apparatus  and  a  constant  speed 
imparted.  Then  let  a  jet  of  air  be  directed, 
as  in  §  396,  against  each  row  of  holes  in 
succession.  It  will  be  found  that  the  musi- 
cal sequence  do,  mi,  sol,  do'  results.  If  the 
speed  of  rotation  is  increased,  each  note 
will  rise  in  pitch,  but  the  sequence  will 
remain  unchanged. 

We  learn,  therefore,  that  the  musi-  FI<J-  36°-   Disk  for  produo- 

cal  sequence  do,  mi,  sol,  do'  consists  of  ^S  ^^cal  sequence  do,  mi, 

*  J  sol,  do' 

notes  whose  vibration  numbers  have  the 

ratios  of  24,  30,  36,  and  48,  that  is,  4,  5,  6,  8,  and  that  this  se- 
quence is  independent  of  the  absolute  vibration  numbers  of  the  tones. 
Furthermore,  when  two  notes  an  octave  apart  are  sounded 
together,  they  form  the  most  harmonious  combination  which  it 
is  possible  to  obtain.  These  characteristics  of  notes  an  octave 
apart  were  recognized  in  the  earliest  times,  long  before  any- 
thing whatever  was  known  about  the  ratio  of  their  vibration 
numbers.  The  preceding  experiment  showed  that  this  ratio 
is  the  simplest  possible,  namely,  24  to  48,  or  1  to  2.  Again, 
the  next  easiest  musical  interval  to  produce  and  the  next 

331 


332  PROPERTIES  OF  MUSICAL  SOUNDS 

most  harmonious  combination  which  can  be  found  corre- 
sponds to  the  two  notes  commonly  designated  as  do,  sol  Our 
experiment  showed  that  this  interval  corresponds  to  the  next 
simplest  possible  vibration  ratio,  namely,  24  to  36,  or  2  to  3. 
When  sol  is  sounded  with  do'  the  vibration  ratio  is  seen  to  be 
36  to  48,  or  3  to  4.  We  see,  therefore,  that  the  three  simplest 
possible  ratios  of  vibration  numbers,  namely,  1  to  2,  2  to  3, 
and  3  to  4,  are  used  in  the  production  of  the  three  notes 
do,  sol,  do'.  Again,  our  experiment  shows  that  another  har- 
monious musical  interval,  do,  mi,  corresponds  to  the  vibration 
ratio  24  to  30,  or  4  to  5.  We  learn,  therefore,  that  harmonious 
musical  intervals  correspond  to  very  simple  vibration  ratios. 

410.  The  major  diatonic  scale.  When  the  three  notes  do, 
mi,  sol,  which,  as  seen  above,  have  the  vibration  ratios  4,. 5,  6, 
are  all  sounded  together,  they  form  a  remarkably  pleasing 
combination  of  tones.  This  combination  was  picked  out  and 
used  very  early  in  the  musical  development  of  the  race.  It  is 
now  known  as  the  major  chord.  The  major  diatonic  scale  is 
built  up  of  three  major  chords.  The  absolute  vibration  number 
taken  as  the  starting  point  is  wholly  immaterial,  but  the  ex- 
planation of  the  origin  of  the  eight  notes  of  the  octave  com- 
monly designated  by  the  letters  C,  I),  E,  F,  G,  A,  B,  C'  may 
be  made  more  simple  if  we  b'egin,  as  above,  with  a  note  whose 
vibration  number  is  24.  If  this  note  is  designated  by  the  letter 
C,  the  two  other  notes  of  the  first  major  chord,  do-mi-sol,  are 
designated  by  E  and  G.  The  second  chord  is  obtained  by 
starting  from  C',  the  octave  of  C,  and  coming  down  in  the 
ratios  6,  5,  4.  The  corresponding  vibration  numbers  are  48, 
40,  and  32,  and  the  corresponding  notes,  known  as  do,  la,  fa, 
are  designated  by  the  letters  C',  A,  and  F.  The  third  chord 
starts  with  G  as  the  first  note  and  runs  up  in  the  ratios  4,  5,  6. 
The  corresponding  notes,  known  as  sol,  si,  re,  have  the  vibration 
numbers  36,  45,  and  54,  and  are  designated  by  the  letters  G,  B, 
and  D'.  It  will  be  seen  that  the  note  D'  does  not  fall  in  the 


VIBRATING  STRINGS  333 

octave  between  C  and  C",  for  its  vibration  number  is  above  48. 
The  note  D  an  octave  below  it  falls  between  C  and  C'  and  has 
a  vibration  number  27.  This  completes  the  eight  notes-  of  the 
diatonic  scale.  The  chord  do-mi-sol  is  called  the  tonic,  sol-si-re 
the  dominant,  and  fa-la-do  the  subdominant.  Below  is  given 
in  tabular  form  the  relations  between  the  notes  of  an  octave. 

Syllables do  re  mi  fa     sol  la  si'  do 

Letters C  -D  E  F      G  A  B  C 

Relative  vibration  numbers    ...    24  27  30  32      36  40  45  48 

Vibration  ratios  in  terms  of  do  .    .      1  f  f  |        f  f  \5-  2 

Absolute  vibration  numbers    .    .      256  288  320  341  £  384  426§  480  512 

Standard  middle  C  forks  made  for  physical  laboratories  all 
have  the  vibration  number  256.  In  the  so-called  international 
pitch  middle  C  has  261  vibrations,  and  in  concert  pitch  274. 

411.  The  even-tempered  scale.    If  G  is  taken  as  do  and  a 
scale  built  up  as  above,  it  will  be  found  that  six  of  the  above 
notes  in  each  octave  can  be  used  in  this  new  key,  but  that  two 
additional  ones  are  required  (see  table  below).    Similarly,  to 
build  up  scales,  as  above,  in  all  the  keys  demanded  by  modern 
music  would  require  about  fifty  notes  in  each  octave.    Hence 
a  compromise  is  made  by  dividing  the  octave  into  twelve 
equal  intervals  represented  by  the  eight  white  and  five  black 
keys  of  a  piano.    How  much  this  so-called  even-tempered  scale 
differs  from  the  ideal,  or  diatonic,  scale  is  shown  below. 

Note                    C     D        E         F        G         A  E  C'  D'  E'         F'  G' 

Diatonic  ....  256    288       320       341J  384  426f  480  512  576  640  682.2  768 

Diatonic  key  of  G 384  432  480  512  576  640  720  768 

Tempered     ...  256   287.4   322.7    341.7  383.8  430.7  4835  512  574.8  645.4  683.4  767.6 

VIBRATING  STRINGS  * 

412.  Laws  of  vibrating  Strings.    Let  two  piano  wires  be  stretched 
over  a  box,  or  a  board  with  pulleys  attached  so  as  to  form  a  sonometer 
(Fig.  361).    Let  the  weights  A  and  B  be  adjusted  until  the  two  wires 
emit  exactly  the  same  note.    The  phenomenon  of  beats  will  make  it 

*  This  discussion  should  be  followed  by  a  laboratory  experiment  on  the  laws  of 
vibrating  strings.  See,  for  example,  Experiment  41  of  the  authors'  manual. 


334 


PEOPEETIES  OF  MUSICAL  SOUNDS 


FIG.  361.   The  sonometer 


possible  to  do  this  with  great  accuracy.  Then  let  the  bridge  D  be  inserted 
exactly  at  the  middle  of  one  of  the  wires,  and  the  two  wires  plucked  in 
succession.  The  interval  will  be  recognized  at  once  as  do,  do'.  Next  let 
the  bridge  be  inserted  so  as  to  make  one  wire  two  thirds  as  long  as  the 
other,  and  let  the  two  be  plucked  again.  The  interval  will  be  recognized 
as  do,  sol. 

Now  it  was  shown 
in  §  409  that  do'  has 
twice  as  many  vibra- 
tions per  second  as 
do,  and  sol  has  three  halves  as  many.  Hence,  since  the  length 
corresponding  to  do'  is  one  half  as  great  as  the  first  length, 
and  that  corresponding  to  sol  two  thirds  as  great,  we  conclude 
from  this  experiment  that,  other  things  being  equal,  the  vibra- 
tion numbers  of  strings  are  inversely  proportional  to  their  lengths. 

Again,  let  the  two  wires  be  tuned  to  unison,  and  then  let  the  weight 
A  be  increased  until  the  pull  which  it  exerts  on  the  wire  is  exactly  four 
times  as  great  as  that  exerted  by  B.  The  note  given  out  by  the  A  wire 
will  again  be  found  to  be  an  octave  above  that  given  out  by  the  B  wire. 

We  learn,  then,  that  the  vibration  numbers  of  similar  strings  of 
equal  length  are  proportional  to  the  square  roots  of  the  tensions. 

413.  Nodes  and  loops  in  vibrating  strings.  Let  a  string  a  meter 

long  be  attached  to  one  of  the  prongs  of  a  large  tuning  fork  which 
makes   in   the   neighbor- 
hood   of    100    vibrations        .Jff—  iiiiiniiiiiiiiiiiiiiiiiiiii|(||||||||[|fnn 

per  second.   Let  the  other 

end  be  attached  as  in  the 

figure,   and   the   fork   set 

into  vibration.  If  the  fork 

is  not  electrically  driven,  which  is  much  to  be  preferred,  it  may  be 

bowed  with  a  violin  bow  or  struck  with  a  soft  mallet.    By  making  the 

tension  of  the  thread,  for  example,  proportional  to  the  numbers  9,  4, 

and  1  it  will  be  found  possible  to  make  it  vibrate  either  as  a  whole,  as 

in  Fig.  362,  or  in  two  or  three  parts  (Fig.  363). 

This  effect  is  due,  as  explained  in  §  408,  to  the  interference 
of  the  direct  and  reflected  waves  sent  down  the  string  from 


FIG.  362.    String  vibrating  as  a  whole 


FUNDAMENTALS  AND  OVERTONES  335 

the  vibrating  fork.    But  we  shall  show  in  the  next  paragraph 
that   in   considering  the   effects   of  the  vibrating    string   on 

the  surrounding  air  we        //      _^L  m ^J^~  — 

shall  make  no  mistake      y 

if  we   think   of  it  as      ?  Vf 

clamped  at  each  node,  FlG-  363-  StrinS  vibrating  in  three 

1  .,  S  segments 

and  as  actually  vibrat- 
ing in  two  or  three  or  four  separate  parts,  as  the  case  may  be. 

FUNDAMENTALS  AND  OVERTONES 

414.  Fundamentals  and  overtones.  If  the  assertion  just 
made  be  correct,  then  a  string  which  has  a  node  in  the  middle 
communicates  twice  as  many  pulses  to  the  air  per  second  as 
the  same  string  when  it  vibrates  as  a  whole.  This  may  be 
conclusively  shown  as  follows : 

Let  the  sonometer  wire  (Fig.  361)  be  plucked  in  the  middle  and  the 
pitch  of  the  corresponding  tone  carefully  noted.  Then  let  the  finger  be 
touched  to  the  middle  of  the  wire,  and  the  latter  plucked  midway  between 
this  point  and  the  end.*  The  octave  of  the  original  note  will  be  dis- 
tinctly heard.  Next  let  the  finger  be  touched  at  a  point  one  third  of  the 
wire  length  from  one  end,  and  the  wire  again  plucked.  The  note  will  be 
recognized  as  sol'.  Since  we  learned  in  §  410  that  sol'  has  three  halves  as 
many  vibrations  as  do',  it  must  have  three  times  as»many  vibrations  as  the 
original  note.  Hence  a  wire  which  is  vibrating  in  three  segments  sends 
.out  three  times  as  many  vibrations  as  when  it  is  vibrating  as  a  whole. 

Now  when  a  wire  is  plucked  in  the  middle  it  vibrates  simply 
as  a  whole,  and  therefore  gives  forth  the  lowest  note  which  it 
is  capable  of  producing.  This  note  is  called  the  fundamental 
of  the  wire.  When  the  wire  is  made  to  vibrate  in  two  parts  it 
gives  forth,  as  has  just  been  shown,  a  note  an  octave  higher 
than  the  fundamental.  This  is  called  the  first  overtone.  When 
the  wire  is  made  to  vibrate  in  three  parts  it  gives  forth  a  note 
corresponding  to  three  times  the  vibration  number  of  the 
fundamental,  namely,  sol'.  This  is  called  the  second  overtone. 

*  It  is  well  to  remove  the  finger  almost  simultaneously  with  the  plucking. 


336  PBOPERTIES  OF  MUSICAL  SOUNDS 

When  the  wire  vibrates  in  four  parts  it  gives  forth  the  third 
overtone,  which  is  a  note  two  octaves  above  the  fundamental. 
The  overtones  of  wires  are  often  called  harmonics.  They  bear 
the  vibration  ratios  2,  3,  4,  5,  6,  7,  etc.,  to  the  fundamental.* 
415.  Simultaneous  production  of  fundamentals  and  overtones. 
We  have  thus  far  produced  overtones  only  by  forcing  the  wire 
to  remain  at  rest  at  certain  properly  chosen  points  during  the 
bowing. 

Now  let  the  wire  be  plucked  at  a  point  one  fourth  of  its  length  from 
one  end,  without  being  touched  in  the  middle.  The  tone  most  distinctly 
heard  will  be  the  fundamental,  but  if  the  wire  is  now  touched  very 
lightly  exactly  in  the  middle, 
the  sound,  instead  of  ceasing 
altogether,  will  continue,  but 
the  note  heard  will  be  an  oc- 
tave  higher  than  the  fnnda-  ^  ^  A  wire  simultaneousiy  emittillg 
mental,  showing  that  in  this  ^  fundamental  and  first  overtone 

case    there    was     superposed 

upon  the  vibration  of  the  wire  as  a  whole  a  vibration  in  two  segments 
also  (Fig.  364).  By  touching  the  wire  in  the  middle  the  vibration  as  a 
whole  was  destroyed,  but  that  in  two  parts  remained.  Let  the  experi- 
ment be  repeated,  with  this  difference,  that  the  wire  is  now  plucked  in 
the  middle  instead  of  one  fourth  its  length  from  one  end.  If  it  is  now 
touched  in  the  middle,  the  sound  will  entirely  cease,  showing  that  when 
a  wire  is  plucked  in  the  middle  there  is  no  first  overtone  superposed 
upon  the  fundamental.  Let  the  wire  be  plucked  again  one  fourth  of  its 
length  from  one  end,  and  careful  attention  given  to  the  compound  note 
emitted.  It  will  be  found  possible  to  recognize  both  the  fundamental 
and  the  first  overtone  sounding  at  the  same  time.  Similarly,  by  plucking 
at  a  point  one  sixth  of  the  length  of  the  wire  from  one  end,  and  then 
touching  it  at  a  point  one  third  of  its  length  from  the  end,  the  second 
overtone  may  be  made  to  appear  distinctly,  and  a  trained  ear  will  detect 
it  in  the  note  given  off  by  the  wire,  even  before  the  fundamental  is 
suppressed  by  touching  at  the  point  indicated. 

*  Some  instruments,  such  as  bells,  can  produce  higher  tones  whose  vibration 
numbers  are  not  exact  multiples  of  the  fundamental.  These  notes  are  still  called 
overtones,  but  they  are  not  called  harmonics,  the  latter  term  being  reserved  for 
the  multiples.  Strings  produce  harmonics  only. 


FUNDAMENTALS  AND  OVERTONES  337 

The  experiments  show,  therefore,  that  in  general  the  note 
emitted  by  a  string  plucked  at  random  is  a  complex  one,  consist- 
ing of  a  fundamental  and  several  overtones,  and  that  just  what 
overtones  are  present  in  a  given  case  depends  on  where  and  how 
the  wire  is  plucked. 

416.  Quality.  Let  the  sonometer  wire  be  plucked  first  in  the  middle 
and  then  close  to  one  end.  The  two  notes  emitted  will  have  exactly  the 
same  pitch,  and  they  may  have  exactly  the  same  loudness,  but  they  will 
be  easily  recognized  as  different  in  respect  to  something  which  we  call 
quality.  The  experiment  of  the  last  paragraph  shows  that  the  real  phys- 
ical difference  in  the  tones  is  a  difference  in  the  sort  of  overtones  which 
are  mixed  with  the  fundamental  in  the  two  cases. 

Again,  let  a  mounted  C"  fork  be  sounded  simultaneously  with  a 
mounted  C  fork.  The  resultant  tone  will  sound  like  a  rich,  full  C, 
which  will  change  into  a  hollow  C  when  the  C'  is  quenched  with  the 
hand. 

Every  one  is  familiar  with  the  fact  that  when  notes  of  the 
same  pitch  and  loudness  are  sounded  upon  a  piano,  a  violin, 
and  a  cornet,  the  three  tones  can  be  readily  distinguished.  The 
last  experiments  suggest  that  the  cause  of  this  difference  lies 
in  the  fact  that  it  is  only  the  fundamental  which  is  the  same 
in  the  three  cases,  while  the  overtones  are  different.  In  other 
words,  the  characteristic  of  a  tone  which  we  call  its  quality  is 
determined  simply  by  the  number  and  prominence  of  the  over- 
tones which  are  present.  If  there  are  few  and  weak  overtones 
present,  while  the  fundamental  is  strong,  the  tone  is,  as  a  rule, 
soft  and  mellow,  as  when  a  sonometer  wire  is  plucked  in  the 
middle,  or  a  closed  organ  pipe  is  blown  gently,  or  a  tuning 
fork  is  struck  with  a  soft  mallet.  The  presence  of  compar- 
atively strong  overtones  up  to  the  fifth  adds  fullness  and 
richness  to  the  resultant  tone.  This  is  illustrated  by  the 
ordinary  tone  from  a  piano,  in  which  several  if  not  all  of  the 
first  five  overtones  have  a  prominent  place.  When  overtones 
higher  than  the  sixth  are  present  a  sharp  metallic  quality 
begins  to  appear.  This  is  illustrated  when  a  tuning  fork  is 


338 


PROPERTIES  OF  MUSICAL  SOUNDS 


struck,  or  a  wire  plucked,  with  a  hard  body.  It  is  in  order  to 
avoid  this  quality  that  the  hammers  which  strike  against  piano 
wires  are  covered  with  felt. 

417.  Analysis  of  tones  by  the  manometric  flame.  A  very 
simple  and  beautiful  way  of  showing  the  complex  character  of 
most  tones  is  furnished  by  the  so-called  manometric  flames. 
This  device  consists  of  the  following  parts :  a  chamber  in  the 
block  B  (Fig.  365),  through  which  gas  is  led  by  way  of  the 


FIG.  365.   Analysis  of  sounds  with  manometric  flames 

tubes  C  and  D  to  the  flame  F ;  a  second  chamber  in  the  block 
A,  separated  from  the  first  chamber  by  an  elastic  diaphragm 
made  of  very  thin  sheet  rubber  or  paper,  and  communicating 
with  the  source  of  sound  through  the  tube  E  and  trumpet  G ; 
and  a  rotating  mirror  M  by  which  the  flame  is  observed.  When 
a  note  is  produced  before  the  mouthpiece  G  the  vibrations  of 
the  diaphragm  produce  variations  in  the  pressure  of  the  gas 
coming  to  the  flame  through  the  chamber  in  5,  so  that  when 
condensations  strike  the  diaphragm  the  height  of  the  flame  is 
increased,  and  when  rarefactions  strike  it  the  height  of  the 


FUNDAMENTALS  AND  OVERTONES 


339 


flame  is  diminished.  If  these  up-and-down  motions  of  the  flame 
are  viewed  in  a  rotating  mirror,  the  longer  and  shorter  images 
of  the  flame,  which  correspond  to  successive  intervals  of  time, 
appear  side  by  side,  as  in  Fig.  366.  If  a  rotating  mirror  is  not 
to  be  had,  a  piece  of  ordinary  mirror  glass  held  in  the  hand 
and  oscillated  back  and  forth  about  a  vertical  axis  will  be 
found  to  give  perfectly  satisfac- 
tory results. 

First  let  the  mirror  be  rotated 
when  no  note  is  sounded  before  the 
mouthpiece.  There  will  be  no  fluc- 
tuations in  the  flame,  and  its  image, 
as  seen  in  the  moving  mirror,  will 
be  a  straight  band,  as  shown  in  2, 
Fig.  366.  Next  let  a  mounted  C  fork 
be  sounded,  or  some  other  simple 
tone  produced  in  front  of  G.  The 
image  in  the  mirror  will  be  that 
shown  in  3.  Then  let  another  fork 
Cf  be  sounded  in  place  of  the  C.  The 
image  will  be  that  shown  in  4.  The 
images  of  the  flame  are  now  twice  as 
close  together  as  before,  since  the 
blows  strike  the  diaphragm  twice  as 
often.  Next  let  the  open  ends  of  the 

resonance  boxes  of  the  two  tuning  forks  C  and  C'  be  held  together  in 
front  of  G.  The  image  of  the  flame  will  be  as  shown  in  5.  If  the  vowel 
o  be  sung  in  the  pitch  Bb  before  the  mouthpiece,  a 'figure  exactly  similar 
to  5  will  be  produced,  thus  showing,  that  this  last  note  is  a  complex, 
consisting  of  a  fundamental  and  its  first  overtone. 

The  proof  that  most  other  tones  are  likewise  complex  lies 
in  the  fact  that  when  analyzed  by  the  manometric  flame  they 
show  figures  not  like  3  and  4,  which  correspond  to  simple 
tones,  but  like  £,  6,  and  7,  which  may  be  produced  by  sounding 
combinations  of  simple  tones.  In  the  figure,  6  is  produced 
by  singing  the  vowel  e  on  Crf ;  7  is  obtained  when  o  is  sung 
on  C". 

t 


FIG. 


Vibration  forms  shown 
by  manometric  flames 


340  PROPERTIES  OF  MUSICAL  SOUNDS 

418.  Helmholtz's  experiment.  If  the  loud  pedal  on  a  piano  is  held 
down  and  the  vowel  sounds  oo,  I,  a,  ah,  e,  sung  loudly  into  the  strings, 
these  vowels  will  be  caught  up  and  returned  by  the  instrument  with 
sufficient  fidelity  to  make  the  effect  almost  uncanny. 

It  was  by  a  method  which  may  be  considered  as  merely  a 
refinement  of  this  experiment  that  Helmholtz  proved  conclu- 
sively that  quality  is  determined  simply  by  the  number  and 
prominence  of  the  overtones  which  are  blended  with  the  fun- 
damental. He  first  constructed  a  large  number  of  resonators, 
like  that  shown  in  Fig.  367,  each  of  which  would  respond  to 
a  note  of  some  particular  pitch.  By  holding  these  resonators 
in  succession  to  his  ear  while  a  musical  note  was  sounding,  he 
picked  out  the  constituents  of  the  note ;  that  is,  he  found  out 
just  what  overtones  were  present  and  what 
were  their  relative  intensities.  Then  he  put 
these  constituents  together  and  reproduced 
the  original  tone.  This  was  done  by  sound- 
ing simultaneously,  with  appropriate  loud- 
ness,  two  or  more,  of  a  whole  series  of  tuning  FlG-  367<  Helm- 
£  ,  ,  .  ,  ,  T  ,,  .,  ,.  ..  -loo  holtz's  resonator 

lorks  which  had  the  vibration  ratios  1,  Z,  o, 

4,  5,  6,  7.  In  this  way  he  succeeded  not  only  in  imitating  the 
qualities  of  different  musical  instruments,  but  even  in  repro- 
ducing the  various  vowel  sounds. 

419.  Sympathetic  vibrations.  Let  two  mounted  tuning  forks  of 
the  same  pitch  be  placed  with  the  open  ends  of  their  resonators  facing 
each  other.  Let  one  be  set  into  Vigorous  vibration  with  a  soft  mallet 
and  then  quickly  quenched  by  grasping  the  prongs  with  the  hand. 
The  other  fork  will  be  found  to  be  sounding  loudly  enough  to  be  heard 
over  a  large  room.  Next  let  a  penny  be  waxed  to  one  prong  of  the  sec- 
ond fork  and  the  experiment  repeated.  When  the  sound  of  the  first 
fork  is  quenched,  no  sound  whatever  will  be  found  to  be  coming  from 
the  second  fork. 

The  experiment  illustrates  the  phenomenon  of  sympathetic 
vibrations  and  shows  what  conditions  are  essential  to  its  appear- 
ance. If  two  bodies  capable  of  emitting  musical  notes  have 


HERMANN  LUDWIG  FERDINAND  VON  HELMHOLTZ  (1821-1894) 

Noted  German  physicist  and  physiologist ;  professor  of  physiology  and  anatomy 
at  Bonn  and  at  Heidelberg  from  1855  to  1871 ;  professor  of  physics  at  Berlin  from 
1871  to  1894 ;  published  in  1847  a  famous  paper  on  the  conservation  of  energy,  which 
was  most  influential  in  establishing  that  doctrine ;  invented  the  ophthalmoscope ; 
discovered  the  physical  significance  of  tone  quality,  and  made  other  contribu- 
tions to  acoustics  and  optics ;  was  preeminent  also  as  a  mathematical  physicist 


FUNDAMENTALS  AND  OVERTONES  341 

exactly  the  same  natural  period  of  vibration,  the  pulses  com- 
municated to  the  air  when  one  alone  is  sounding,  beat  upon  the 
second  at  intervals  which  correspond  exactly  to  its  own  'natu- 
ral period.  Each  pulse  therefore  adds  its  effect  to  that  of  the 
preceding  pulses,  and  though  the  effect  due  to  a  single  pulse 
is  very  slight,  a  great  number  of  such  pulses  produce  a  large 
resultant  effect.  In  the  same  way  a  large  number^of  very  feeble 
pulls  may  set  a  heavy  pendulum  into  vibrations  of  considerable 
amplitude  if  the  pulls  come  at  intervals  exactly  equal  to  the 
natural  period  of  the  pendulum.  On  the  other  hand,  if  the 
two  sounding  bodies  have  even  a  slight  difference  of  period, 
the  effect  of  the  first  pulses  is  neutralized  by  the  effect  of  suc- 
ceeding pulses  as  soon  as  the  two  bodies,  on  account  of  their 
difference  in  period,  get  to  swinging  in  opposite  directions. 

Let  notes  of  different  pitches  be  sung  into  a  piano  when  the  dampers 
are  lifted.  The  wire  which  has  the  pitch  of  the  note  sounded  will  in 
every  case  respond.  Sing  a  little  off  the  key  and  the  response  will  cease. 

420.  Sympathetic  vibrations  produced  by  overtones.    It  is 

not  essential,  in  order  that  a  body  may  be  set  into  sympathetic 
vibrations,  that  it  have  the  same  pitch  as  the  sounding  body, 
provided  its  pitch  corresponds  exactly  with  the  pitch  of  one  of 
the  overtones  of  that  body. 

Thus,  if  the  damper  is  lifted  from  the  C  string  of  a  piano  and  the 
octave  below,  Cv  is  sounded  loudly,  C  will  be  heard  to  sound  after  Cl  has 
been  quenched  by  the  damper.  In  this  case  it  is  the  first  overtone  of  Cl 
which  is  in  exact  tune  with  C,  and  which  therefore  sets  it  into  sympa- 
thetic vibration.  Again,  if  the  damper  is  lifted  from  the  G  string  while 
C1  is  sounded,  this  note  will  be  found  to  be  set  into  vibration  by  the 
second  overtone  of  Cr  A  still  more  interesting  case  is  obtained  by  remov- 
ing the  damper  from  E  while  Cl  is  sounded.  When  Cx  is  quenched,  the 
note  which  is  heard  is  not  E,  but  an  octave  above  E]  that  is,  E'.  This  is 
because  there  is  no  overtone  of  Cl  which  corresponds  to  the  vibration  of 
E;  but  the  fourth  overtone  of  Cv  which  has  five  times  the  vibration  num- 
ber of  Cj,  corresponds  exactly  to  the  vibration  number  of  E',  the  first 
overtone  of  E.  Hence  E  is  set  into  vibration  not  as  a  whole  but  in  halves. 


342 


PEOPEKTIES  OF  MUSICAL  SOUNDS 


421.  Physical  significance  of  harmony  and  of  discord.  Let  two 

pieces  of  glass  tubing  about  an  inch  in  diameter  and  a  foot  and  a  halt 
long  be  supported  vertically,  as  shown  in  Fig.  368.  Let  two  gas  jets, 
made  by  drawing  down  pieces  of  one-fourth  inch  glass  tubing  until,  with 
full  gas  pressure,  the  flame  is  about  an  inch  long,  be  thrust  inside  these 
tubes  to  a  height  of  about  three  or  four  inches  from  the  bottom.  Let 
the  gas  be  turned  down  until  the  tubes  begin  to  sing.  Without  attempt- 
ing to  discuss  the  part  which  the  flame  plays  in  the  production  of  the 
sound,  we  wisK  simply  to  call  attention  to  the  fact 
that  the  two  tones  are  either  quite  in  unison,  or  so 
near  it  that  but  a  few  beats  are  produced  per  second. 
Now  let  the  length  of  one  of  the  tubes  be  slightly 
increased  by  slipping  the  paper  cylinder  S  up  over 
its  end.  The  number  of  beats  will  be  rapidly  in- 
creased until  they  will  become  indistinguishable  as 
separate  beats  and  will  merge  into  a  jarring,  grat- 
ing discord. 

The  experiment  teaches  that  discord  is  sim- 
ply a  phenomenon  of  beats.  If  the  vibration 
numbers  do  not  differ  by  more  than  five  or 
six,  that  is,  if  there  are  not  more  than  five 
or  six  beats  per  second,  the  effect  is  not  par- 
ticularly unpleasant.  From  this  point  on, 
however,  as  the  difference  in  the  vibration 

numbers,   and   therefore  in   the  number  of 

FIG.  368.  Illustrat- 

beats  per  second,  increases,  the  unpleasant-  ing  the  production 
ness  increases,  and  becomes  worst  at  a  differ-  of  discords 
ence  of  about  thirty.  Thus  the  notes  B  and 
(7',  which  differ  by  about  thirty-two  beats  per  second,  produce 
about  the  worst  possible  discord.  When  the  vibration  numbers 
differ  by  as  much  as  seventy,  which  is  about  the  difference 
between  C  and  E,  the  effect  is  again  pleasing  or  harmonious. 
Moreover,  in  order  that  two  notes  may  harmonize  well,  it  is 
necessary  not  only  that  the  notes  themselves  shall  not  pro- 
duce an  unpleasant  number  of  beats,  but  also  that  such  beats 
shall  not  arise  from  their  overtones.  Thus  C  and  B  are  very 


FUNDAMENTALS  AND  OVERTONES  343 

discordant,  although  they  differ  by  a  large  number  of  vibra- 
tions per  second.  The  discord  in  this  case  arises  between  B 
and  C'r  the  first  overtone  of  C.  , 

Again,  there  are  certain  classes  of  instruments,  of  which  bells 
are  a  striking  example,  which  produce  insufferable  discords 
when  even  such  notes  as  do,  sol,  do1,  are  sounded  simultaneously 
upon  them.  This  is  because  these  instruments,  unlike  strings 
and  pipes,  have  overtones  which  are  not  harmonics,  that  is, 
which  are  not  multiples  of  the  fundamental ;  and  these  over- 
tones produce  beats  either  among  themselves  or  with  one  of 
the  fundamentals.  It  is  for  this  reason  that  in  playing  chimes 
the  bells  are  struck  in  succession,  not  simultaneously. 

QUESTIONS  AND  PROBLEMS 

1.  At  what  point  must  the  G1  string  be  pressed  by  the  finger  of  the 
violinist  in  order  to  produce  the  note  CY? 

2.  If  one  wire  has  twice  the  length  of  another  and  is  stretched  by  four 
times  the  stretching  force,  how  will  their  vibration  numbers  compare  ? 

3.  A  wire  gives  out  the  note  G.    What  is  its  fourth  overtone?. 

4.  What  is  the  wave  length  of  middle  C  when  the  speed  of  sound  is 
1152  ft.  per  second? 

5.  What  is  the  pitch  of  a  note  whose  wave  length  is  5.4  in.,  the  speed 
being  1152  ft.  per  second? 

6.  If  middle  C  Had  300  vibrations  per  second^  how  many  vibrations 
would  F  and  A  have  ? 

7.  What  is  the  fourth  overtone  of  C?  the  fifth  overtone? 

8.  A. wire  gives  out  the  note  C  when  the  tension  on  it  is  4  kg.    What 
tension  will  be  required  to  give  out  the  note  (7? 

9.  A  wire  50  cm.  long  gives  out  400  vibrations  per  second.    How 
many  vibrations  will  it  give  when  the  length  is  reduced  to  10  cm.  ?  What 
syllable  will  represent  this  note  if  do  represents  the  first  note  ? 

10.  There  are  seven  octaves  and  two  notes  on  an  ordinary  piano,  the 
lowest  note  being  A±  and  the  highest  one  C"".   If  the  vibration  number 
of  the  lowest  note  is  27,  find  the  vibration  number  of  the  highest. 

11.  Find  the  wave  length  of  the  lowest  note  on  the  piano;  the  wave 
length  of  the  highest  note.    (Take  the  speed  of  sound  in  air  as  1130  ft- 
per  second.) 

12.  A  violin  string  is  commonly  bowed  about  one  seventh  of  its  length 
from  one  end.    Why  is  this  better  than  bowing  in  the  middle  ? 


344 


PEOPEETIES  OF  MUSICAL  SOUNDS 


WIND  INSTRUMENTS 

422.  Fundamentals  of  closed  pipes.   Let  a  tightly  fitting  rubber 

stopper  be  inserted  in  a  glass  tube  a  (Fig.  369),  eight  or  ten  inches  long 
and  about  three  fourths  of  an  inch  in  diameter.  Let  the  stopper  be 
pushed  along  the  tube  until,  when  a  vibrating  C'  fork  is  held  before  the 
mouth,  resonance  is  obtained  as  in  §  402.  (The  length  will  be  six  or 
seven  inches.)  Then  let  the  fork  be  removed  and  a  stream  of  air  blown 
across  the  mouth  of  the  tube  through  a  piece  of 
tubing  b,  flattened  at  one  end  as  in  the  figure.*  " 

The  pipe  will  be  found  to  emit  strongly  the 
note  of  the  fork. 

In  every  case  it  is  found  that  a  note 
which  a  pipe  may  be  made  to  emit  is 
always  a  note  to  which  it  is  able  to  re- 
spond when  used  as  a  resonator.  Since, 
in  §  403,  the  best  resonance  was  found 
when  the  wave  length  given  out  by  the 
fork  was  four  times  the  length  of  the 
pipe,  we  learn  that  when  a  current  of  air 
is  suitably  directed  across  the  mouth  of  a  closed  pipe,  it  will  emit 
a  note  which  has  a  wave  length  four  times  the  length  of  the  pipe. 
This  note  is  called  the  fundamental  of  the  pipe.  It  is  the  lowest 
note  which  the  pipe  can  be  made  to  produce. 

423.  Fundamentals  of  open  pipes.    Since  we  found  in  §  404 
that  the  lowest  note  to  which  a  pipe  open  at  the  lower  end  can 
respond  is  one  the  wave  length  of  which  is  twice  the  pipe 
length,  we  infer  that  an  open  pipe  when  suitably  blown  ought 
to  emit  a  note  the  wave  length  of  which  is  twice  the  pipe  length. 
This  means  that  if  the  same  pipe  is  blown  first  when  closed  at 
the  lower  end  and  then  when  open,  the  first  note  ought  to  be 
an  octave  lower  than  the  second. 

*  If  the  arrangement  of  Fig.  369  is  not  at  hand,  simply  hlow  with  the  lips  across 
the  edge  of  a  piece  of  ordinary  glass  tubing  within  which  a  rubber  stopper  may 
be  pushed  back  and  forth. 


FIG.  369.  Musical  notes 
from  pipes 


WIND  INSTRUMENTS  345 

Let  the  pipe  a  (Fig.  369)  be  closed  at  the  bottom  with  the  hand  and 
blown ;  then  let  the  hand  be  removed  and  the  operation  repeated.  The 
second  note  will  indeed  be  found  to  be  an  octave  higher  than  the  first. 

We  learn,  therefore,  that  the  fundamental  of  an  open  pipe 
has  a  wave  length  equal  to  twice  the  pipe  length. 

424.  Overtones  in  pipes.  It  was  found  in  §  403  that  there 
are  a  whole  series  of  pipe  lengths  which  respond  to  a  given 
fork,  and  that  these  lengths  bear  to  the  wave  length  of  the 
fork  the  ratios  \,  f ,  |,  etc.  This  is  equivalent  to  saying  that 
a  closed  pipe  of  fixed  length  can  respond  to  a  whole  series  of 
notes  whose  vibration  numbers  have  the  ratios  1,  3,  5,  7,  etc. 
Similarly,  in  §  404,  we  found  that  in  the  case  of  an  open  pipe 
the  series  of  pipe  lengths  which  will  respond  to  a  given  fork 
bear  to  the  wave  length  of  the  fork  the  ratios  J-,  f ,  f ,  |-,  etc. 
This  again  is  equivalent  to  saying  that  an  open  pipe  can  re- 
spond to  a  series  of  notes  whose  vibration  numbers  have  the 
ratios  1,  2,  3,  4,  5,  etc.  Hence  we  infer  that  it  ought  to  be 
possible  to  cause  both  open  and  closed  pipes  to  emit  notes  of 
higher  pitch  than  their  fundamentals,  that  is,  overtones,  and 
that  the  first  overtone  of  an  open  pipe  should  have  twice  the 
rate  of  vibration  of  the  fundamental,  that  is,  that  it  should  be 
do',  the  fundamental  being  considered  as  do ;  that  the  second 
overtone  should  vibrate  three  times  as  fast  as  the  fundamental, 
that  is,  it  should  be  sol' ;  that  the  third  overtone  should  vibrate 
four  times  as  fast,  that  is,  it  should  be  do" ';  that  the  fourth 
overtone  should  vibrate  five  times  as  fast,  that  is,  it  should  be 
mi",  etc.  In  the  case  of  the  closed  pipe,  however,  the  first 
overtone  should  have  a  vibration  rate  three  times  that  of  the 
fundamental,  that  is,  it  should  be  sol' ;  the  second  overtone 
should  vibrate  five  times  as  fast,  that  is,  it  should  be  mi",  etc. 
In  other  words,  while  an  open  pipe  ought  to  give  forth  all  the 
harmonics,  both  odd  and  even,  a  closed  pipe  ought  to  pro- 
duce the  odd  harmonics,  but  be  entirely  incapable  of  producing 
the  even  ones. 


346  PROPERTIES  OF  MUSICAL  SOUNDS 

Let  the  pipe  of  Fig.  369  be  blown  so  as  to  produce  the  fundamental 
when  the  lower  end  is  open.  Then  let  the  strength  of  the  air  blast  be 
increased.  The  note  will  be  found  to  spring  to  do'  .  By  blowing  still 
harder  it  will  spring  to  sol',  and  a  still  further  increase  will  probably 
bring  out  do''.  When  the  lower  end  is  closed,  however,  the  first  overtone 
will  be  found  to  be  sol'  and  the  next  one  mi",  just  as  our  theory  demands. 

425.  Mechanism  of  emission  of  notes  by  pipes.  A  musical 
note  is  produced  by  blowing  across  the  mouth  of  a  pipe  because 
the  jet  of  air  vibrates  back  and  forth  across  the  lip  in  a  period 
which  is  determined  wholly  by  the  natural  resonance  period  of 
the  pipe.  Thus  suppose  that  the  jet  a  (Fig.  370)  first  strikes 
just  inside  the  edge  or  lip  of  the  pipe.  A  condensational  pulse 
starts  down  the  pipe.  When  it  returns  to  the 
mouth  after  reflection  at  the  closed  end  it 
pushes  the  jet  outside  the  lip.  This  starts  a 
rarefaction  down  the  pipe,  which,  after  return 
from  the  lower  end,  pulls  the  jet  in  again. 
There  are  thus  sent  out  into  the  room  regu- 
larly timed  puffs  the  period  of  which  is  con- 
trolled by  the  reflected  pulses  coming  back 


from  the  lower  end;  that  is,  by  the  natural    FlG;370- 

.-,/.',         .  ing  air  jet 

resonance  period  of  the  pipe. 

By  blowing  more  violently  it  is  possible  to  create,  by  virtue 
of  the  friction  of  the  walls,  so  great  and  so  sudden  a  compres- 
sion in  -the  mouth  of  the  pipe  that  the  jet  is  forced  out  over 
the  edge  before  the  return  of  the  first  reflected  pulse.  In  this 
case  no  note  will  be  produced  unless  the  blowing  is  of  just 
the  right  intensity  to  cause  the  jet  to  swing  out  in  the  period 
corresponding  to  an  overtone.  In  this  case  the  reflected  pulses 
will  return  from  the  end  at  just  the  right  intervals  to  keep  the 
jet  swinging  in  this  period.  This  shows  why  a  current  of  a  par- 
ticular intensity  is  required  to  start  any  particular  overtone. 

Another  way  of  looking  at  the  matter  is  to  think  of  the 
pipe  as  being  filled  up  with  air  until  the  pressure  within  it  is 


WIND  INSTRUMENTS 


347 


great  enough  to  force  the  jet  outside  the  lip,  upon  which  a 
period  of  discharge  follows,  to  be  succeeded  in  turn  by 
another  period  of  charge.  These  periods  are  controlled  by  the 
length  of  the  pipe  and  the  violence  of  the  blowing  precisely 
as  described  above. 

With  open  pipes  the  situation  is  in  no  way  different  save 
that  the  reflection  of  a  condensation  as  a  rarefaction  at  the 
lower  end  makes  the  natural  period  twice  as  high,  since  the 
pipe  length  is  now  one-half  wave  length  instead  of  one-fourth 
wave  length  (see  §  404). 

426.  Vibrating  air-jet  instruments.   The  mechanism  of  the  production 
of  musical  tones  by  the  ordinary  organ  pipe,  the  flute,  the  fife,  the 
piccolo,  and  all  whistles  is  essentially  the  same  as  in 

the  case  of  the  pipe  of  Fig.  370.  In  all  these  instru- 
ments an  air  jet  is  made  to  play  across  the  edge  of  an 
opening  in  an  air  chamber,  and  the  reflected  pulses 
returning  from  the  other  end  of  the  chamber  cause  it 
to  vibrate  back  and  forth,  first  into  the  chamber  and 
then  out  again.  In  this  way  a  series  of  regularly 
timed  puffs  of  air  is  made  to  pass  from  the  instru- 
ment to  the  ear  of  the  observer  precisely  as  in  the 
case  of  the  rotating  disk  of  §  396.  The  air  chamber 
may  be  either  open  or  closed  at  the  remote  end.  In 
the  flute  it  is  open,  in  whistles  it  is  usually  closed, 
and  in  organ  pipes  it  may  be  either  open  or  closed. 
Fig.  371  shows  a  cross  section  of  two  types  of  organ 
pipes.  The  jet  of  air  from  S  vibrates  across  the  lip 
L  in  obedience  to  the  pressure  exerted  on  it  by  waves 
reflected  from  0.  Pipe  organs  are  provided  with  a 
different  pipe  for  each  note,'  but  the  flute,  piccolo,  or 
fife  is 'made  to  produce  a  whole  series  of  notes,  either  by  blowing  over- 
tones or  by  opening  holes  in  the  tube,  an  operation  which  is  equivalent 
to  cutting  the  tube  off  at  the  hole. 

427.  Vibrating  reed  instruments.    In  reed  instruments  the  vibrating 
air  jet  is  replaced  by  a  vibrating  reed  or  tongue  which  opens  and  closes, 
at  absolutely  regular  intervals,  an  opening  against  which  the  performer 
is  directing  a  current  of  air.   In  the  clarinet,  the  oboe,  the  bassoon,  etc., 
the  reed  is  placed  at  the  upper  end  of  the  tube  (see  I,  Fig.  372),  and  the 


0 

L 

H 

ir 

\ 

1        U 

FIG.  371.   Organ 
pipes 

350  PROPERTIES  OF  MUSICAL  SOUNDS 

QUESTIONS  AND  PROBLEMS 

1.  What  will  be  the  relative  lengths  of  a  series  of  organ  pipes  which 
produce  the  eight  notes  of  a  diatonic  scale  ? 

2.  What  must  be  the  length  of  a  closed  organ  pipe  which  produces 
the  note  E'l    (Take  the  speed  of  sound  as  340  m.  per  second.) 

3.  Will  the  pitch  of  a  pipe  organ  be  the  same  in  summer  as  on  a 
cold  day  in  winter  ?    What  could  cause  a  difference  ? 

4.  What  is  the  first  overtone  which  can  be  produced  in  an  open  G 
organ  pipe  ? 

5.  What  is>  the  first  overtone  which  can  be  produced  by  a  closed 
C  organ  pipe? 

6.  Explain  how  an  instrument  like  the  bugle,  which  has  an  air 
column  of  unchanging  length,  may  be  made  to  produce  several  notes 
of  different  pitch. 

7.  When  water  is  poured  into  a  deep  bottle,  why  does  the  pitch  of 
the  sound  rise  as  the  bottle  fills  ? 

8.  Why  is  the  quality. of  an  open  organ  pipe  different  from  that  of 
a  closed  organ  pipe  ? 

9.  The  velocity  of  sound  in  hydrogen  is  about  four  times  as  great 
as  it  is  in  air.    If  a  C  pipe  is  blown  with  hydrogen,  what  will  be  the 
pitch  of  the  note  emitted  ? 

10.  What  effect  will  be  produced  on  a  phonograph  record  made  with 
the  instrument  of  Fig.  377  if  the  loudness  of  a  note  is  increased  ?  if 
the  pitch  is  lowered  an  octave? 


CHAPTER  XVIII 


I 


NATURE  AND  PROPAGATION  OF  LIGHT 
TRANSMISSION  OF  LIGHT 

430.  Speed  of  light.  Before  the  year  1675  light  was  thought 
to  pass  instantaneously  from  the  source  to  the  observer.  In 
that  year,  however,  Olaf  Roemer,  a  young  Danish  astron- 
omer, made  the  following  observations.  He  had  observed 
accurately  the  instant  at  which  one  of  Jupiter's  satellites  M 
(Fig.  378)  passed  into 
Jupiter's  shadow  when 
the  earth  was  at  E, 
and  predicted,  from  the 
known  mean  time  be- 
tween such  eclipses,  the 
exact  instant  at  which 
a  given  eclipse  should 
occur  six  months  later 
when  the  earth  was 
atJZ'.  It  actually  took  FIG.  378.  Illustrating  Roemer's  determination 

^  of  the  velocity  of  light 

place    16   minutes    36 

seconds  (996  seconds)  later.  He  concluded  that  the  996 
seconds'  delay  represented  the  time  required  for  light  to 
travel  across  the  earth's  orbit,  a  distance  known  to  be  about 
180,000,000  miles.  The  most  precise  of  modern  determinations 
of  the  speed  of  light  are  made  by  laboratory  methods.  The 
generally  accepted  value,  that  of  Michelson,  of  The  University 
of  Chicago,  is  299,860  kilometers  per  second.  It  is  sufficiently 
correct  to  remember.it  as  300,000  kilometers,  or  186,000  miles. 

351 


352        NATURE  AND  PROPAGATION  OF  LIGHT 

Though  this  speed  would  carry  light  around  the  earth  7J  times 
in  a  second,  yet  it  is  so  small  in  comparison  with  interstellar 
distances  that  the  light  which  is  now  reaching  the  earth  from 
the  nearest  fixed  star,  Alpha  Centauri,  started  4.4  years 
ago.  If  an  observer  on  the  pole  star  had  a  telescope  powerful 
enough  to  enable  him  to  see  events  on  the  earth,  he  would  not 
see  the  battle  of  Gettysburg  (which  occurred  in  July,  1863) 
until  January,  1918. 

Both  Foucault  in  France  and  Michelson  in  America  have 
measured  directly  the  velocity  of  light  in  water  and  have 
found  it  to  be  only  three  fourths  as  great  as  in  air.  It  will  be 
shown  later  that  in  all  transparent  liquids  and  solids  it  is  less 
than  it  is  in  air. 

431.  Reflection  of  light.*  Let  a  beam  of  sunlight  be  admitted  to 
a  darkened  room  through  a  narrow  slit.  The  straight  path  of  the  beam 
will  be  rendered  visible  by  the  brightly  illumined  dust  particles  sus- 
pended in  the  air.  Let  the  beam  fall  on  the  surface  of  a  mirror.  Its 
direction  will  be  seen  to  be  sharply 
changed,  as  shown  in  Fig.  379.  Let 
the  mirror  be  held  so  that  it  is  per- 
pendicular to  the  beam.  The  beam  will 
be  seen  to  be  reflected  directly  back 
on  itself.  Let  the  mirror  be  turned 
through  an  angle  of  45°.  The  reflected 
beam  will  move  through  90°. 

The  experiment  shows  roughly, 
therefore,  that  the  angle  /OP,  be-  FlG> 
tween  the  incident  beam  and  the 
normal  to  the  mirror,  is  equal  to  the  angle  FOR  between  the  re- 
flected beam  and  the  normal  to  the  mirror.  The  first  angle,  10 P, 
is  called  the  angle  of  incidence,  and  the  second,  POR,  the  angle  of 
reflection.  Hence  the  law  of  the  reflection  of  light  may  be  stated 
thus  :  The  angle  of  reflection  is  equal  to  the  angle  of  incidence. 

*An  exact  laboratory  experiment  on  the  law  of  reflection  should  either  precede 
or  follow  this  discussion.  See,  for  example,  Experiment  42  of  the  authors'  manual. 


A.  A.  MICHELSON,  CHICAGO 

Distinguished  for  extraordinarily  accu- 
rate experimental  researches  in  light. 
First  American  scientist  to  receive  the 
Nobel  prize 


LORD  RAYLEIGH  (ENGLAND) 

Distinguished  for  the  discovery  of  argon, 
for  very  accurate  determinations  in  elec- 
tricity and  sound  and  for  profound  theo- 
retical studies 


HENRY  A.  ROWLAND,  JOHNS  HOPKINS 
Distinguished  for  the  invention  of  the 


SIR  WILLIAM  CROOKES,  LONDON 

Distinguished  for  his  pioneer  work  (1875) 

concave  grating  and  for  epoch-making      in  the  study  and  interpretation  of  cath- 
studies  in  heat  and  electricity  ode  rays  (pp.  418  and  423) 

A  GROUP  OF  MODERN  PHYSICISTS 


TRANSMISSION  OF  LIGHT  353 

432.  Diffusion   of   light.    In  the  last  experiment  the  light  was 
reflected  by  a  very  smooth  plane  surface.    Let  the  beam  be  now  allowed 
to  fall  upon  a  rough  surface  like  that  of  a  sheet  of  unglazed  white  paper. 
No  reflected  beam  will  be  seen ;  but,  instead,  the  whole  room  will  be 
brightened  appreciably,  so  that  the  outline  of  objects  before  invisible 
may  be  plainly  distinguished. 

The  beam  has  evidently  been  scattered  in  all  directions  by 
the  innumerable  little  reflecting  surfaces  of  which  the  sur- 
face of  the  paper  is  composed.  The  effect  will  be  much  more 
noticeable  if  the 
beam  is  allowed 
to  fall  alternately 
on  a  piece  of  dead 
black  cloth  and  on 
the  white  paper. 

The  light  is  largely  absorbed  by  the  cloth,  while  it  is  scattered 
or  diffusely  reflected  by  the  paper.  The  difference  between  a 
smooth  reflector  and  a  rough  one  is  illustrated  in  greatly 
magnified  form  in  Fig.  380. 

433.  Visibility  of  nonluminous  bodies.    Every  one  is  familiar 
with  the  fact  that  certain  classes  of  bodies,  such  as  the  sun,  a 
gas  flame,  etc.,  are  self-luminous,  that  is,  visible  on  their  own 
account ;  while  other  bodies,  like  books,  chairs,  tables,  etc.,  can 
be  seen  only  when  they  are  in  the  presence  of  luminous  bodies. 
The  above  experiment  shows  how  such  nonluminous,  diffusing 
bodies  become  visible  in  the  presence  of  luminous  bodies.   For, 
since  a  diffusing  surface  scatters  in  all  directions  the  light  which 
falls  upon  it,  each  small  element  of  such  a  surface  is  sending 
out  light  in  a  great  many  directions,  in  much  the  same  way  in 
which  each  point  on  .a  luminous  surface  is  sending  out  light  in 
all  directions.    Hence  'we  always  see  the  outline  of  a  diffusing 
surface  as  we  do  that  of  an  emitting  surface,  no  matter  where 
the  eye  is  placed.    On  the  other  hand,  when  light  comes  to 
the  eye  from  a  polished  reflecting  surface,  since  the  form  of 


354       NATURE  AND  PEOPAGATION  OF  LIGHT 


the  beam  is  wholly  undisturbed  by  the  reflection,  we  see  the 
outline  not  of  the  mirror  but  rather  of  the  source  from  which 
the  light  came  to  the  mirror,  whether  this  source  is  itself  self- 
luminous  or  is  only  acting,  because  of  its  light-scattering 
power,  like  a  self-luminous  source.  Points  on  the  mirror  which 
are  not  in  line  with  this  source  can  send  no  light  whatever  to 
the  eye.  Hence  the  mirror  itself  must  be  invisible.  The  fact 
that  one  often  runs  into  a  large  mirror  or  plate-glass  window 
is  sufficient  confirmation  of  the  truth  of  the  statement  that 
neither  a  perfect  reflector  nor  a  perfectly  transparent  body  is 
itself  visible.  All  bodies  other  than  self-luminous  ones  are 
visible  only  by  the  light  which  they  diffuse.  Black  bodies  send 
no  light  to  the  eye,  but  their  outlines  can  be  distinguished  by 
the  light  which  comes  from  the  background.  Any  object  which 
can  be  seen,  therefore,  may  be  regarded  as  itself  sending  rays 
to  the  eye ;  that  is,  it  may  be  treated  as  a  luminous  body. 

434.  Refraction.  Let  a  narrow  beam  of  sunlight  be  allowed  to  fall 
on  a  thick  glass  plate  with  a  polished  front  and  whitened  back*  (Fig.  381). 
It  will  be  seen  to  split  into  a  re- 
flected and  a  transmitted  portion. 
The  transmitted  portion  will  be 
seen  to  be  bent  toward  the  per- 
pendicular OP  drawn  into  the 
glass.  Upon  emergence  into  the 
air  it  will  be  bent  again,  but  this 
time  away  from  the  perpendicu- 
lar O'P'  drawn  into  the  air.  Let 
the  incident  beam  strike  the  sur- 
face at  different  angles.  It  will  be 
seen  that  the  greater  the  angle  of  in- 
cidence the  greater  the  bending.  At 
normal  incidence  there  will  be  no  bending  at  all..  If  the  upper  and  lower 
"faces  of  the  glass  are  parallel,  the  bending  at  the  two  faces  will  always  be 
the  same,  so  that  the  emergent  beam  is  parallel  to  the  incident  beam. 

*  All  of  these  experiments  on  reflection  and  refraction  may  be  done  effectively 
and  conveniently  by  using  disks  of  glass,  like  those  used  with  the  Hartl  Optical 
Disk,  through  which  the  beam  can  be  traced. 


FIG.  381.    Path  of  a  ray  through  a  me- 
dium bounded  by  parallel  faces 


TRANSMISSION  OF  LIGHT 


355 


Similar  experiments  made  with  other  substances  have 
rought  out  the  general  law  that  whenever  light  travels  olliquely 
>*om  one  medium  into  another  in  which  the  speed  is  less,  it  is  bent 
)ward  the  perpendicular,  and  when  it  passes  from  one  medium  to 
nother  in  which  the  speed  is  greater, 
'  is  bent  away  from  the  perpendic- 
lar  drawn  into  the  second  medium. 


FIG.  382.  Rays  coming  from  a 
source  I  under  water  to  the 
boundary  between  air  and  water 
at  different  angles  of  incidence 


435.  Total  reflection ;  critical 
ngle.  Since  rays  emerging  from 

medium  like  water  into  one  of 
388  density  like  air  are  always 
>ent  from  the  perpendicular  (see 
TA,  ImB,  etc.,  Fig.  382),  it  is  clear 
hat  if  the  angle  of  incidence  on 
he  under  surface  of  the  water  is 

lade  larger  and  larger,  a  point  must  be  reached  at  which  the 
efracted  ray  is  parallel  to  the  surface  (see  InC,  Fig.  382).  It 
3  interesting  to  inquire  what  will  happen  to  a  ray  lo  which 
trikes  the  surface  at  a  still  greater  angle  of  incidence  loP1. 
t  will  not  be  unnatural  to  suppose  that  since  the  ray  nC  just 
grazed  the  surface,  the  ray  lo  will 
!iot  be  able  to  emerge  at  all.  The 
olio  wing  experiment  will  show  that 
his  is  indeed  the  case. 


FIG.  383.  Transmission  and 
reflection  of  light  at  surface 
AB  of  a  right-angled  prism 


Let  a  prism  with  three  polished  edges, 
polished  front,  and  a  whitened  back  be 
eld  in  the  path  of  a  narrow  beam  of  sun- 
ight,  as  shown  in  Fig.  383.  If  the  angle 
>f  incidence  loP  is  small,  the  beam  will 
livide  at  0  into  a  reflected  and  a  trans- 
nitted  portion,  the  former  going  to  £', 

he  latter  to  S  (neglect  fee  color  for  the  present).  Let  the  prism  be 
otated  slowly  in  the  direction  of  the  arrow.  A  point  will  be  reached 
it  which  the  transmitted  beam  disappears  completely,  while  at  the  same 
ime  the  spot  at  S'  shows  an  appreciable  increase  in  brightness.  Since 
t 


356       NATURE  AND  PROPAGATION  OF  LIGHT 

the  transmitted  ray  OS  has  totally  disappeared,  the  whole  of  the  lighi 
incident  at  0  must  be  in  the  reflected  beam.  The  angle  of  incidence 
IOP  at  which  this  occurs  is  called  the  critical  angle.  This  angle  foi 
crown  glass  is  42.5°,  for  water  48.5°,  for  diamond  23.7°. 

We  learn  then  that  when  a  ray  of  light  traveling  in  an% 
medium  meets  another  in  which  the  speed  is  greater,  it  is  totally 
reflected  if  the  angle  of  incidence  is  greater  than  a  certain  anglt 
called  the  critical  angle. 

QUESTIONS  AND  PROBLEMS 

1.  Sirius,  the  brightest  star,  is  about  52,000,000,000,000  miles  away 
If  it  were  suddenly  annihilated,  how  long  would  it  shine  on  for  us  ? 

2.  In  Fig.  384  the  portion  acdb  of  the  shadow  is  called  the  umbra 
the  portions  aec  and  bdf  the  penumbra.  What 

kind  of  a  source  has  no  penumbra? 

3.  If  the  opaque  body  in  Fig.  384  is  moved 
nearer  to  the  screen  ef,  how  does  the  penumbra 
change  ? 

4.  The  sun  is  much  larger  than  the  earth. 
Draw  a  diagram  showing  the  shape  of  the 
earth's  umbra  and  penumbra. 

5.  The  diameter  of  the  moon  is  2000  miles, 
that  of  the  sun  860,000  miles,  and  the  sun  is 

93,000,000  miles  away.  What  is  the  length  of 

,       o  FIG.  384.    Shadow  froc 

the  moon's  umbra  ? 

a  broad  source 

6.  Will  it  ever  be  possible  for  the  moon  to 

totally  eclipse  the  sun  from  the  whole  of  the  earth's  surface  at  once 

7.  If  the  distance  from  the  center  of  the  earth  to  the  center  of  th 
moon  were  exactly  equal  to  the  length  of  the  moon's  umbra,  over  ho\ 
wide  a  strip  on  the  earth's  surface  would  the  sun  be  totally  eclipsed  a 
any  one  time? 

8.  Why  is  a  room  with  white  walls  much  lighter  than  a  simila 
room  with  black  walls? 

9.  If  the  word  "white"  be  painted  with  white  paint  (or  whitin; 
moistened  with  alcohol)  across  the  face  of  a  mirror  and  held  in  the  pat] 
of  a  beam  of  sunlight  in  a  darkened  room,  in  the  middle  of  the  spot  01 
the  wall  which  receives  the  reflected  beam  the  word  "  white  "  will  appea 
in  black  letters.    Explain. 

10.  Look  at  the  reflected  image  of  an  electric-light  filament  in 
piece  of  red  glass.    Why  are  there  two  images,  one  red  and  one  white 


TRANSMISSION  OF  LIGHT 


357 


11.  The  earth  reflects  sixteen  times  as  much  light  to  the  moon  as  the 
aoon  does  to  the  earth.   Trace  from  the  sun  to  the  eye  of  the  observer  the 
(tght  by  which  he  is  able  to  see  the  dark  part  of  the  new  moon.   Why 
ian  we  not  see  the  dark  part  of  a  third- 

;  uarter  moon  ? 

12.  If  a  penny  is  placed  in  the  bottom 
f  a  vessel  in  such  a  position  that  the  edge 
ust  hides  it  from  view  (Fig.  385),  it  will 

Become  visible  as  soon  as  water  is  poured  ^      OQK 

Into  the  vessel.    Explain. 

13.  A  stick  held  in  water  appears  bent,  as  shown  in  Fig.  386.  Explain. 

14.  Should  a  man  who  wishes  to  spear  a  fish  aim  high  or  low? 

15.  A  glass  prism  placed  in  the  position  shown  in  Fig.  387  is  the 
aost  perfect  reflector  known.   Why  is  it  better  than  an  ordinary  mirror  ? 


FIG. 


FIG.  387 


FIG.  388 


16.  What  is  the  principal  reflecting  medium  in  an  ordinary  mirror? 

17.  Explain  why  a  straight  wire  seen  obliquely  through  a  piece  of 
l^lass  appears  broken,  as  in  Fig.  388. 

18.  In  what  direction  must   a  fish   look    in 
!»rder  to  see  the  setting  sun?    (See  Fig.  389.) 


FIG.  389.  To  an  eye  under  water  all  external  ob- 
jects appear  to  lie  within  a  cone  whose  angle  is  97° 


FIG.  390.  Luxfer 

prism  glass 


19.  Fig.  390  represents  a  section  of  a  plate  of  Luxfer  prism  glass. 
Explain  why  glass  of  this  sort  is  so  much  more  efficient  than  ordinary 
vindow  glass  in  illuminating  the  rears  of  dark  stores  on  the  ground 
l©or  in  narrow  streets. 


358       NATURE  AND  PROPAGATION  OF  LIGHT 

THE  NATURE  OF  LIGHT 

436.  The  corpuscular  theory  of  light.    All  of  the  propertied 
of  light  which  have  so  far  been  discussed  are  perhaps  most 
easily  accounted  for  on  the  hypothesis  that  light  consists  o'i 
streams  of  very  minute  particles,  or  corpuscles,  projected  wit! 
the  enormous  velocity  of  300,000  kilometers  per  second  fror, 
all  luminous  bodies.    The  facts  of  straight-line  propagatio 
and  reflection  are  exactly  as  we  should  expect  them  to  be  ij 
this  were  the  nature  of  light.    The  facts  of  refraction  can  alsc 
be  accounted  for,   although   somewhat  less   simply,  on   thi* 
hypothesis.    As  a  matter  of  fact,  this  theory  of  the  nature  pj! 
light,  known  as  the   corpuscular  theory,  was   the   one  moslj 
generally  accepted  up  to  about  1800. 

437.  The  wave  theory  of  light.    A  rival  hypothesis,  whicll 
was  first  completely  formulated  by  the  great  Dutch  physicisii 
Huygens  (1629-1695),  regarded  light,  like  sound,  as  aforn 
of  wave  motion.    This  hypothesis  met  at  the  start  with  twc 
very  serious  difficulties.    In  the  first  place,  light,  unlike  sound 
not  only  travels  with   perfect  readiness    through   the    bes' 
vacuum  which  can  be  obtained  with  an  air  pump,  but  it  travel; 
without  any  apparent  difficulty  through  the  great  interstella: 
spaces  which  are  probably  infinitely  better  vacua  than  can  b< 
obtained  by  artificial  means.    If,  therefore,  light  is  a  wav< 
motion,  it  must  be  a  wave  motion  of  some  medium  which  fill; 
all  space  and  yet  which  does  not  hinder  the  motion  of  th<| 
stars  and  planets.    Huygens  assumed  such  a  medium  to  exist 
and  called  it  the  ether. 

The  second  difficulty  in  the  way  of  the  wave  theory  of  ligh 
was  that  it  seemed  to  fail  to  account  for  the  fact  of  straight 
line  propagation.  Sound  waves,  water  waves,  and  all  othe 
forms  of  waves  with  which  we  are  most  familiar  bend  readib 
around  corners,  while  light  apparently  does  not.  It  was  thi 
difficulty  chiefly  which  led  many  of  the  most  famous  of  th< 


i    i 


i_ 


CHRISTIAN  HUYGENS  (1629-1695) 

Great  Dutch  physicist,  mathematician,  and  astronomer;  dis- 
covered the  rings  of  Saturn;  made  important  improvements  in 
the  telescope ;  invented  the  pendulum  clock  (1656) ;  developed 
with  marvelous  insight  the  wave  theory  of  light ;  discovered  in 
1690  the  "polarization"  of  light.  (The  fact  of  double  refraction 
was  discovered  by  Erasmus  Bartholinus  in  1669,  but  Huygens 
first  noticed  the  polarization  of  the  doubly  refracted  beams,  and 
offered  an  explanation  of  double  refraction  from  the  standpoint 
of  the  wave  theory) 


THE  NATURE  OF  LIGHT 


359 


early  philosophers,  including  the  great  Sir  Isaac  Newton,  to 
reject  the  wave  theory  and  to  support  the  projected-particle 
theory.  Within  the  last  hundred  years,  however,  this  difficulty 
has  been  completely  removed,  and  in  addition  other  properties 
of  light  have  been  discovered  for  which  the  wave  theory  offers 
the  only  satisfactory  explanation.  The  most  important  of  these 
properties  will  be  treated  in  the  next  paragraph. 

438.  Interference  of  light.  Let  two  pieces  of  plate  glass  about 
half  an  inch  wide  and  four  or  five  inches  long  be  separated  at  one  end  by 
a  thin  sheet  of  paper  in  the  manner  shown  in  Fig.  391,  while  the  other 
end  is  clamped  or  held  firmly  together,  so  that  a  very  thin  wedge  of 
air  exists  between  the  plates.  Let  a  piece 
of  asbestos  or  blotting  paper  be  soaked  in 
a  solution  of  common  salt  (sodium  chlo- 
ride) and  placed  over  the  tube  of  a  Bunsen 
burner  so  as  to  touch  the  flame  in  the 
manner  shown.  The  flame  will  be  colored 
a  bright  yellow  by  the  sodium  in  the  salt. 
When  the  eye  looks  at  the  reflection  of 
'the  flame  from  the  glass  surfaces,  a  series 
of  fine  black  and  yellow  lines  will  be  seen 
to  cross  the  plate. 


Paper 


FIG.  391.   Interference  of 
light  waves 


The  wave  theory  offers  the  fol- 
lowing explanation  of  these  effects. 
Each  point  of  the  flame  sends  out 
light  waves  which  travel  to  the  glass 
plate  and  are  in  part  reflected  and  in 
part  transmitted  at  all  the  surfaces  of  the  glass,  that  is,  at  A'B', 
at  AB,  at  CD,  and  at  C'D'  (Fig.  391).  We  will  consider,  how- 
ever, only  those  reflections  which  take  place  at  the  two  faces  of 
the  air  wedge,  namely,  at  AB  and  CD.  Let  Fig.  392  represent  a 
greatly  magnified  section  of  these  two  surfaces.  Let  the  wavy 
line  as  represent  a  light  wave  reflected  from  the  surface  AB 
at  the  point  a,-  and  returning  thence  to  the  eye.  Let  the  dotted 
wavy  line  ir  represent  a  lightwave  reflected  from  the  surface  CD 


360       NATURE  AND  PROPAGATION  OF  LIGHT 


CA 


at  the  point  i,  and  returning  thence  to  the  eye.  Similarly,  let 
all  the  continuous  wavy  lines  of  the  figure  represent  light 
waves  reflected  from  different  points  on  AB  to  the  eye,  and 
let  all  the  dotted  wavy  lines  represent  waves  reflected  from 
corresponding  points  on  CD  to  the  eye.  Now,  in  precisely  the 
same  way  in  which  two  trains  of  sound  waves  from  two  tun- 
ing forks  were  found,  in  the  experiment  illustrating  beats  (see 
§  407),  to  interfere 
with  each  other  so 
as  to  produce  silence 
whenever  the  two 
waves  'corresponded 
to  motions  of  the  air 
particles  in  opposite 
directions,  so  in  this 
experiment  the  two 
sets  of  light  waves 
from  AB  and  CD 
interfere  with  each 
other  so  as  to  pro- 
duce darkness  wher- 
ever these  two  waves 
correspond  to  mo- 
tions of  the  light-transmitting  medium  in  opposite  directions. 
The  dark  bands,  then,  of  our  experiment  are  simply  the  places 
at  which  the  two  beams  reflected  from  the  two  surfaces  of 
the  air  film  neutralize  or  destroy  each  other,  while  the  light 
bands  correspond  to  the  places  at  which  the  two  beams  reen- 
force  each  other  and  thus  produce  illumination  of  double 
intensity. 

Now  the  condition  for  destructive  interference  is  obviously 
that  the  wave  which  passes  thro-  gh  the  film  and  is  reflected 
from  any  point  on  CD,  such,  for  example,  as  i,  shall  return  to 
AB,  after  its  double  passage  through  the  film,  in  the  condition 


interference 

reenforcement 

interference 

reenforcement 

interference 

reenforcement 

interference 

reenforcement 


D 


FIG.  392.   Explanation  of  formation  of  dark  and 
light  bands  by  interference  of  light  waves 


THE  NATURE  OF  LIGHT  361 

or  phase  of  vibration  which  is  exactly  opposite  to  that  of  the 
wave  which  is  being  reflected  at  that  instant  from  the  corre- 
sponding point  on  AB,  namely  from  a.  If  this  condition  occurs 
at  a,  it  must  occur  again  at  a  point  c  enough  farther  down  the 
wedge  to  make  the  double  path  through  the  film  just  one  wave 
length  more,  and  again  at  e  where  the  double  path  is  two  wave 
lengths  more,  etc.  In  other  words,  the  dark  bands  ought  to 
follow  one  another  at  equal  intervals  down  the  wedge,  pre- 
cisely as  we  observed  them  to  do.  Between  each  two  successive 
points  of  interference  there  must,  of  course,  be  a  point  like  5, 
d,  f,  or  h,  at  which  the  waves  reflected  from  the  two  surfaces 
unite  in  like  phases  and  therefore  reenforce  each  other.  This 
phenomenon  of  the  interference  of  light  is  met  with  in  many 
different  forms,  and  in  every  case  the  wave  theory  furnishes  at 
once  a  wholly  satisfactory  explanation  of  the  observed  effects ; 
while  the  corpuscular  theory,  on  the  other  hand,  is  unable  to 
account  for  any  of  these  interference  effects  without  the  most 
fantastic  and  violent  assumptions.  Hence  the  corpuscular  theory 
is  now  practically  abandoned,  and  light  is  universally  regarded 
by  physicists  as  a  form  of  ivave  motion. 

439.  The  ether.  We  have  already  indicated  that  if  the  wave 
theory  is  to  be  accepted,  we  must  conceive,  with  Huygens,  that 
all  space  is  filled  with  a  medium,  called  the  ether,  in  which  the 
waves  can  travel.  This  medium  cannot  be  like  any  of  the 
ordinary  forms  of  matter ;  for  if  any  of  these  forms  existed  in 
interplanetary  space,  the  planets  and  the  other  heavenly  bodies 
would  certainly  be  retarded  in  their  motions.  As  a  matter  of 
fact,  in  all  the  hundreds  of  years  during  which  astronomers 
have  been  making  accurate  observations  of  the  motions  of  heav- 
enly bodies  no  such  retardation  has  ever  been  observed.  The 
medium  which  transmits  light  waves  must  therefore  have  a 
density  which  is  infinitelyrsmall  even  in  comparison  with  that 
of  our  lightest  gases.  The  existence  of  such  a  medium  is  now 
universally  assumed~by  physicists. 


362       NATURE  AND  PROPAGATION  OF  LIGHT 

Further,  in  order  to  account  for  the  transmission  of  light 
through  transparent  bodies,  it  is  necessary  to  assume  that  the 
ether  penetrates  not  only  all  interstellar  spaces  but  all  inter- 
molecular  spaces  as  well. 

440.  Wave  length  of  yellow  light.    Although  light,  like  sound,  is  a 
form  of  wave  motion,  light  waves  differ  frpm  sound  waves  in  several 
important  respects.    In  the  first  place,  an  analysis  of  the  preceding  experi- 
ment, which  seems  to  establish  so  conclusively  the  correctness  of  the 
wave  theory,  shows  that  the  wave  length  of  the  light  waves  used  in 
that  experiment  is  extremely  minute  in  comparison  with  that  of  ordi- 
nary sound  waves.   Thus,  suppose  the  air  wedge  was  10  centimeters  long, 
and  that  the  upper  edges  of  the  glass  strips  were  in  contact,  while  the 
lower  edges  were  held  apart  by  a  sheet  of  paper  .03  millimeter  thick. 
Suppose,  further,  that  the  black  bands  were  found  to  be  1  millimeter 
apart.    Now,  since  the  air  wedge  is  100  millimeters  long,  the  difference 
in  its  thickness  at  two  points  such  as  a  and  c  (see  Fig.  392),  1  milli- 
meter apart,  must  be  .01  of  its  thickness  at  the  base,  that  is,  y^  of 
.03  millimeter,  or  .0003  millimeter.    Since  it  is  the  double  path  through 
the  air  wedge  at  c  which  must  be  exactly  one  wave  length  longer  than  the 
double  path  at  a  (see  §  438),  we  see  that  the  difference  in  the  thicknesses 
of  the  wedge  at  c  and  at  a  must  be  ^  wave  length.    Hence  the  wave  length 
of  yellow  light  must  be  2  x  .0003  =  .0006  millimeter.    Careful  measure- 
ments by  better  methods  give  .000589  millimeter  as  the  correct  value. 

The  number  of  vibrations  per  second  made  by  the  little  particles  which 
send  out  the  light  waves  may  be  found,  as  in  the  case  of  sound,  by 
dividing  the  velocity  by  the  wave  length.  Since  the  velocity  of  light  is 
30,000,000,000  centimeters  per  second  and  the  wave  length  is  .00006 
centimeter,  the  number  of  vibrations  per  second  of  the  particles  which 
emit  yellow  light  has  the  enormous  value  500,000,000,000,000. 

441.  Wave  theory  explanation  of  refraction.  Let  one  look  ver- 
tically down  upon  a  glass  or  tall  jar  full  of  water  and  place  his  finger 
on  the  side  of  the  glass  at  the  point  at  which  the  bottom  appears  to  be, 
as  seen  through  the  water  (Fig.  393).    In  every  case  it  will  be  found 
that  the  point  touched  by  the  finger  will  be  about  one  fourth  of  the 
depth  of  the  water  above  the  bottom. 

According  to  the  wave  theory  this  effect  is  due  to  the  fact 
that  the  speed  of  light  is  less  in  water  than  in  air.  Thus 
consider  a  wave  which  originates  at  any  point  P  (Fig.  394) 


THE  NATURE  OF  LIGHT 


363 


beneath  a  surface  of  water  and  spreads  from  that  point  with 

equal  speed  in  all  directions.   At  the  instant  at  which  the  front 

of  this  wave  first  touches  the  surface  mn  it  will, 

of  course,  be  of  spherical  form,  having  P  as  its 

center.    Let  aob  be  a  section  of  this  sphere.   An 

instant  later,  if  the  speed  had  not  changed  in 

passing  into  air,  the  wave  would  have  still  had 

P  as  its  center,  and  its  form  would  have  coin- 

cided with  the  dotted  line  co^d,  so  drawn  that 

ac,  oo^  and  bd  are  all  equal.    But  if  the  velocity 

in  air  is  greater  than  in  water,  then  at  the 

instant  considered  the  disturbance  will  have 

reached  some  point  02  instead  of  o^  and  hence 

the  emerging  wave  will  actually  have  the  form 

of  the  heavy  line  co2d  instead  of  the  dotted  line 


FIG.  393.  Appar- 
ent elevation   of 
the  bottom  of  a 
body  of  water 


co±d.  Now  this  wave  cozd  is  more  curved  than 

the  old  wave  aob,  and  hence  it  has  its  center  at  some  point  P' 

above  P.    In  other  words,  the  wave  has  bulged  upward  in 

passing  from  water  into  air.    Therefore,  when  a  section  of 

this  wave  enters  the  eye  at  E,  the  wave  appears  to  originate 

not  at  P  but  at  P',  for  the  light 

actually  comes  to  the  eye  from 

P'  as  a  center  rather  than  from 

P.     We    conclude,     therefore, 

that    if  light   travels    slower   in 

water  than  in  air,  all  objects  be- 

neath the  surface  of  water  ought 

to  appear  nearer  to  the  eye  than 

they  actually  are.    This  is  pre- 

cisely what  we  found  to  be  the 

case  in  our  experiment. 


FIG.  394.    Representing  a  wave 
emerging  from  water  into  air 


Furthermore,  since  when  the  eye  is  in  any  position  other 
than  E,  for  example  E1,  the  light  travels  to  it  over  the  broken 
path  PSE',  the  construction  shows  that  light  is  always  bent 


364       NATUKE  AND  PBOPAGATION  OF  LIGHT 

away  from  the  perpendicular  when  it  passes  obliquely  into  a 
medium  in  which  the  speed  is  greater.  If  it  had  passed  into 
a  medium  of  less  speed,  the  point  P  would  evidently  have 
appeared  depressed  below  its  natural  position,  and  hence  the 
oblique  rays  would  have  appeared  to  be  bent  toward  the 
perpendicular,  as  we  found  in  §  434  to  be  the  case. 

442.  Ratio  of  the  speeds  of  light  in  air  and  water.  The  last 
experiment  not  only  indicates  qualitatively  that  the  speed  of 
light  is  greater  in  air  than  in  water,  but  it  furnishes  a  simple 
means  of  determining  the  precise  ratio  of  the  two  speeds.  Thus 
in  Fig.  394  the  line  oo2  represents  just  how  far  the  wave  travels 
in  air  while  it  is  traveling  the  distance  ac  (=  oo^)  in  water. 

r\r\ 

Hence  — -  is  the  ratio  of  the  speeds  of  light  in  air  and  in  water. 

.°°i 

Now  it  may  be  shown  that  when  the  arc  cod  is  small,  a  con- 
dition which  is  in  general  realized  in  experimental  work, 

— 3  is  equal  to  — -..*  But  in  our  experiment  we  found  that 
oo1  oP> 

the  bottom  was  raised  one  fourth  of  the  depth ;  that  is,  that 

— -  =  — .  We  conclude,  therefore,  that  light  travels  three 
oP  o 

fourths  as  fast  in  water  as  in  air. 

The  fact  that  the  value  of  this  ratio,  as  determined  by  this 
indirect  method,  is  exactly  the  same  as  that  found  by  Foucault 
and  Michelson,  by  direct  measurement  (§  430),  furnishes  one 
of  the  strongest  proofs  of  the  correctness  of  the  wave  theory. 

*  For  as  the  wave  goes  through  the  surface  at  o  its  curvature  changes  from 
that  of  an  arc  which  has  its  center  at  P  to  that  of  an  arc  which  has  its  center 
at  P'.  Now  the  curvatures  of  these  two  arcs  are  hy  definition  the  reciprocals  of 

their  radii ;  that  is,  they  are  —  and  — — ,  respectively.    The  naturalness  of  this 

definition  may  be  seen  from  the  fact  that  if  at  a  point  like  o  one  arc  is  curving 
twice  as  fast  as  another,  it  is  evident  that  this  means  merely  that  its  center  is  but 
half  as  far  away ;  that  is,  the  ciu-vatures  of  the  two  arcs  are  inversely  proportional 
to  their  radii.  But  so  long  as  the  arcs  co\d  and  co2d  are  very  short,  their  curvatures 
are  also  directly  proportional  to  ooj  and  oo2 ;  that  is,  they  are  proportional  to  the 
amounts  by  which  these  two  curved  lines  depart  from  the  straight  line  cod. 

Hence  oo1/oot=oP'/oP. 


* 
THE  NATURE  OF  LIGHT  365 

443.  Index  of  refraction.    The  ratio  of  the  speed  of  light  in 
air  to  its  speed  in  any  medium  is  called  the  index  of  refraction 
of  that  medium.    It  is  evident  that  the  method  employed  in 
the  last  paragraph  for  determining  the  index  of  refraction  of 
water  can  be  easily  applied  to  any  transparent  medium  whether 
liquid  or  solid.  The  refractive  indices  of  some  of  the  commoner 
substances  are  as  follows : 

Water 1.33  Crown  glass 1.53 

Alcohol 1.36  Flint  glass 1.67 

Turpentine 1.47  Diamond 2.47 

444.  Light  waves  are  transverse.   Thus  far  we  have  discov- 
ered but  two  differences  between  light  waves  and  sound  waves ; 
namely,  the  former  are  disturbances  in  the  ether  and  are  of 
very  short  wave  length,  while  the  latter  are  disturbances  in 
ordinary  matter  and  are  of  relatively  great  wave  length.   There 
exists,,  however,  a  further  radical  difference  which  follows 
from  a 'capital  discovery  p  ^ 
madebyHuygensinthe 

year  1690.    It  is  this: 
While  sound  waves  con-         ' 
sist,  as  we  have  already 
seen,  of  longitudinal  vi- 
brations  of  the  particles 

of  the  transmitting  me- 
•j.        .  ,  -,     ,    .         M       ,-  FIG.  395.   Tranverse  waves  passing 

dmm,  that  is,  vibrations  through  ^ 

back  and  forth  in  the 

line  of  propagation  of  the  wave,  light  waves  are  like  the  water 
waves  of  Fig.  349,  p.  319,  in  that  they  consist  of  transverse 
vibrations,  that  is,  vibrations  of  the  medium  at  right  angles  to 
the  direction  of  the  line  of  propagation. 

In  order  to  appreciate  the  difference  between  the  behavior 
of  waves  of  these  two  types  under  certain  conditions,  conceive 
of  transverse  waves  in  a  rope  to  be  made  to  pass  through  two 
gratings  in  succession,  as  in  Fig.  395.  So  long  as  the  slits  in 


366       NATURE  AND  PROPAGATION  OF  LIGHT 


both  gratings  are  parallel  to  the  plane  of  vibration  of  the 
hand,  as  in  Fig.  395,  (1),  the  waves  can  pass  through  them 
with  perfect  ease;  but  if  the  slits  in  the  first  grating  P  are 
parallel  to  the  direction  of  vibration,  while  those  of  the  second 
grating  Q  are  turned  at  right  angles  to  this  direction,  as  in 
Fig.  395,  (2),  it  is  evident  that  the  waves  will  pass  readily 
through  P,  but  will  be  stopped  completely  by  $,  as  shown 
in  the  figure.  In  other  words,  these 
gratings  P  and  Q  will  let  through 

only  such   vibrations   as    are  paral- 

,   ,    ,       ,-,        -,.       ,.  £    ,-,     .       ,.,  FIG.  396.   Tourmaline  tongs 

lei  to  the  direction  01  their  slits. 

If,  on  the  other  hand,  a  longitudinal  instead  of  a  transverse 
wave  —  such,  for  example,  as  a  sound  wave  —  had  approached 
such  a  grating,  it  would  have  been  as  much  transmitted  in 
one  position  of  the  grating  as  in  another,  since  a  to-and-fro 
motion  of  the  particles  can  evidently  pass  through  the  slits 
with  exactly  the  same  ease,  no  matter  how  they  are  turned. 

Now  two  crystals  of  tourmaline  are  found  to  behave  with 
respect  to  light  waves  precisely  as  the  two  gratings  behave 
with  respect  to  the 
waves  on  the  rope. 

Let  one  such  crys- 
tal a  (Fig.  396)  be  held 
in  front  of  a  small  hole 
in  a  screen  through 
which  a  beam  of  sun- 
light is  passing  to  a 
neighboring  wall ;  or, 

if  the  sun  is  not  shining,  simply  let  the  crystal  be  held  between  the  eye 
and  a  source  of  light.  The  light  will  be  readily  transmitted,  although 
somewhat  diminished  in  intensity.  Then  let  a  second  crystal  b  be  held 
in  line  with  the  first.  The  light  will  still  be  transmitted,  provided  the 
axes  of  the  crystals  are  parallel,  as  shown  in  Fig.  397.  When,  However, 
one  of  the  crystals  is  rotated  in  its  ring  through  90°  (Fig.  398),  the 
light  is  cut  off.  This  shows  that  a  crystal  of  tourmaline  is  capable  of 
transmitting  only  light  which  is  vibrating  in  one  particular  plane. 


FIG.  397.    Light  pass- 
ing through  tourmaline 
crystals 


FIG.  398.    Light  cut  off 

by  crossed   tourmaline 

crystals 


THE  NATUKE  OF  LIGHT 


367 


From  this  experiment,  therefore,  we  are  forced  to  conclude 
that  light  waves  are  transverse  rather  than  longitudinal  vibrations. 

The  above  experiment  illustrates  what  is  technically  known 
as  the  polarization  of  light,  and  the  beam  which,  after  passage 
through  a,  is  unable  to  pass  through  b  if  the  axes  of  a  and 
b  are  crossed,  is  known  as  a  polarized  beam.  It  is,  then,  the 
phenomenon  of  the  polarization  of  light  upon  which  we  base 
the  conclusion  that  light  waves  are  transverse. 

445.  Intensity  of  light.  Let  four  candles  be  set  as  close  together 
as  possible  in  such  a  position  B  as  to  cast  upon  a  white  screen  C,  placed 
in  a  well-darkened  room,  a  shadow  of  an  opaque  object  0  (Fig.  399).  Let 
one  single  candle  be  placed  in  a 
position  A  such  that  it  will  cast 
another  shadow  of  0  upon  the 
screen.  Since  light  from  A  falls 
on  the  shadow  cast  by  B,  and 
light  from  B  falls  on  the  shadow 
cast  by  A,  it  is  clear  that  the 
two  shadows  will  appear  equally  dark  only  when  light  of  equal  in- 
tensity falls  on  each ;  that  is,  when  A  and  B  produce  equal  illumina- 
tion upon  the  screen.  Let  the  positions  of  A  and  B  be  shifted  until  this 
condition  is  fulfilled.  Then  let  the  distances  from  B  to  C  and  from  A 


FIG.  399.    Rumford's  photometer 


FIG.  400.    Proof  of  law  of  inverse  squares 

to  C  be  measured.  If  all  five  candles  are  burning  with  flames  of  the 
same  size,  the  first  distance  will  be  found  to  be  just  twice  as  great  as 
the  second.  Hence  the  illumination  produced  upon  the  screen  by  each 
one  of  the  candles  at  B  is  but  one  fourth  as  great  as  that  produced  on 
the  screen  by  one  candle  at  A,  one  half  as  far  away. 

The  above  is  the  direct  experimental  proof  that  the  intensity  of 
light  varies  inversely  as  the  square  of  the  distance  from  the  source. 

The  theoretical  proof  of  the  law  is  furnished  at  once  by 
Fig.  400,  for  since  all  the  light  which  falls  from  the  candle  L 


368       NATUKE  AND  PEOPAGATION  OF  LIGHT 

on  A  is  spread  over  four  times  as  large  an  area  when  it  reaches 
B,  twice  as  far  away,  and  over  nine  times  as  large  an  area 
when  it  reaches  (7,  three  times  as  far  away,  obviously  the  in- 
tensities at  B  and  at  C  can  be  but  one  fourth  and  one  ninth 
as  great  as  at  A. 

The  above  method  of  comparing  experimentally  the  inten- 
sities of  two  lights  was  first  used  by  Count  Rumford.  The 
arrangement  is  therefore  called  the  Rumford  photometer  (light 
measurer). 

446.  Candle  power.  The  last  experiment  furnishes  a  method 
of  comparing  the  light-emitting  powers  of  various  sources  of 
light.  For  example,  suppose  that  the  four  candles  at  B  are 
replaced  by  a  gas  flame,  and  that  for  the  condition  of  equal 
illumination  upon  the  screen  the  two  distances  BC  and  AC  are 
the  same  as  above,  namely  2  to  1.  We  should  then  know  that 
the  gas  flame,  which  is  able  to  produce  the  same  illumination 
at  a  distance  of  two  feet  as  a  candle  at  a  distance  of  one  foot, 
has  a  light-emitting  power  equal  to  four  candles.  In  general, 
then,  the  candle  power  of  any  two  sources  which  produce  equal 
illumination  on  a  given  screen  are  directly  proportional  to  the 
squares  of  the  distances  of  the  sources  from  the  screen. 

It  is  customary  to  express  the  intensities  of  all  sources  of 
light  in  terms  of  candle  power,  one  candle  power  being  defined 
as  the  amount  of  light  emitted  by  a  sperm  candle  |  inch  in 
diameter  and  burning  120  grains  (7.776  grams)  per  hour.  The 
candle  power  of  an  ordinary  gas  flame  burning  5  cubic  feet 
per  hour  is  from  16  to  25,  depending  on  the  quality  of  the  gas. 
A  Welsbach  lamp  burning  3  cubic  feet  per  hour  has  a  candle 
power  of  from  50  to  100.  Most  incandescent  electric  lamps 
which  are  used  for  domestic  purposes  are  of  16  candle  power. 
The  average  arc  light  has  a  candle  power  of  about  500, 
although  when  measured  in  the  direction  of  greatest  intensity 
the  illuminating  power  may  be  as  great  as  that  of  1000  or 
1200  candles. 


THE  NATURE  OF  LIGHT  369 

447.  Bunsen's  photometer.  Let  a  drop  of  oil  or  melted  paraffin  be 
placed  in  the  middle  of  a  sheet  of  unglazed  white  paper  to  render  it 
translucent.  Let  the  paper  be  held  near  a  window  and  the  side  away 
from  the  window  observed.  The  oiled  spot  will  appear  lighter  than  the 
remainder  of  the  paper.  Then  let  the  paper  be  held  so  that  the  side 
nearest  the  window  may  be  seen.  The  oiled  spot  will  appear  darker 
than  the  rest  of  the  paper.  We  learn,  therefore,  that  when  the  paper  is 
viewed  from  the  side  of  greater  illumination,  the  oiled  spot  appears  dark ; 
but  when  it  is  viewed  from  (he  side  of  lesser  illumination,  the  spot  appears  light. 
If,  then,  the  two  sides  of  the  paper  are  equally  illuminated,  the  spot 
ought  to  be  of  the  same  brightness 
when  viewed  from  either  side.  Let  the 
room  be  darkened  and  the  oiled  paper 
placed  between  two  gas  flames,  two  elec- 
tric lights,  or  any  two  equal  sources  of 
light.  It  will  be  observed  that  when  FIG.  401.  Bunsen's  photometer 
the  paper  is  held  closer  to  one  than  the 

other,  the  spot  will  appear  dark  when  viewed  from  the  side  next  the 
closer  light ;  but  if  it  is  then  moved  until  it  is  nearer  the  other  source, 
the  spot  will  change  from  dark  to  light  when  viewed  always  from  the 
same  side.  It  is  always  possible  to  find  some  position  for  the  oiled 
paper  at  which  the  spot  either  disappears  altogether  or  at  least  appears 
the  same  when  viewed  from  either  side.  This  is  the  position  at  which 
the  illuminations  from  the  two  sources  are  equal.  Hence,  to  find  the 
candle  power  of  any  unknown  source  it  is  only  necessary  to  set  up 
a  candle  on  one  side  and  the  unknown  source  on  the  other,  as  in 
Fig.  401,  and  to  move  the  spot  A  to  the  position  of  equal  illumination. 
The  candle  power  of  the  unknown  source  at  C  will  then  be  the  square 
of  the  distance  from  C  to  A,  divided  by  the  square  of  the  distance 
from  B  to  A. 

This  arrangement  is  known  as  the  Runsen  photometer. 

QUESTIONS  AND  PROBLEMS 

1.  What    is    the    speed    of    light    in    water?    (Index    of    refraction 
is  1.33.) 

2.  Will  a  beam  of  light  going  from  water  into  flint  glass  be  bent 
toward  or  away  from  the  perpendicular  drawn  into  the  glass  ? 

3.  If  the  wedge-shaped  film  of  air  in  Fig.  391  were  replaced  by  water, 
would  the  distance  between  successive  fringes  be  greater  or  less  than  in 
air?   Why? 

t 


370        NATURE  AND  PROPAGATION  OF  LIGHT 

4.  When  light  passes  obliquely  from  air  into  carbon  bisulphide  it  is 
bent  more  than  when  it  passes  from  air  into  water  at  the  same  angle. 
Is  the  speed  of  light  in  carbon  bisulphide  greater  or  less  than  in  water? 

5.  Does  a  man  above  the  surface  of  water  appear  to  a  fish  below  it 
farther  from  or  nearer  to  the  surface  than  he  actually  is  ? 

6.  How  far  from  a  screen  must  a  4-candle-power  light  be  placed  to 
give  the  same  illumination  as  a  16-candle-power  electric  light  3  m.  away  ? 

7.  A  Bunsen  photometer  placed  between  an  arc  light  and  an  incan- 
descent light  of  32  candle  power  is  equally  illuminated  on  both  sides 
when  it  is  10  ft.  from  the  incandescent  light  and  36  ft.  from  the  arc 
light.   What  is  the  candle  power  of  the  arc  ? 

8.  A  5-candle-power  and  a  30-candle-power  source  of  light  are  2  m. 
apart.    Where  must  the  oiled  disk  of  a  Bunsen  photometer  be  placed  in 
order  to  be  equally  illuminated  on  the  two  sides  by  them  ? 

9.  If  the  sun  were  at  the  distance  of  the  moon  from  the  earth,  in- 
stead of  at  its  present  distance,  how  much  stronger  would  sunlight  be 
than  at  present?   The  moon  is  240,000  mi.  and  the  sun  93,000,000  mi. 
from  the  earth. 

10.  If  a  gas  flame  is  300  cm.  from  the  screen  of  a  Rumford  photom- 
eter, and  a  standard  candle  50  cm.  away  gives  a  shadow  of  equal  in- 
tensity, what  is  the  candle  power  of  the  gas  flame  ? 


CHAPTER  XIX 


IMAGE  FORMATION 
IMAGES  FORMED  BY  LENSES 

448.  Focal  length  of  a  convex  lens.  Let  a  convex  lens  be  held 
in  the  path  of  a  beam  of  sunlight  which  enters  a  darkened  room,  where 
it  is  made  plainly  visible  by  means  of  chalk  dust  or  smoke.  The  beam 
will  be  found  to  converge  to  a  focus  F,  as  shown  in  Fig.  402. 

The  explanation  is  as  follows:  The  waves  from  the  sun 
or  any  distant  object  are  without  any  appreciable  curvature 
when  they  strike  they  lens, 
that  is,  they  are  so  called 
plane  waves  (see  Fig.  402). 
Since  the  speed  of  light  is 
less  in  glass  than  in  air, 
the  central  portion  of  these 


FIG.  402.   Principal  focus  F  and  focal 
length  CF  of  a  convex  lens 


waves  is  retarded  more  than  the  outer  portions  in  passing 
through  the  lens.  Hence  on  emerging  from  the  lens  the 
waves  are  concave  instead  of  plane,  and  close  in  to  a  center 
or  focus  at  F. 

A  second  way  of  looking  at  the  phenomenon  is  to  think  of 
the  "  rays  "  which  strike  the  lens  as  being  bernt  by  it,  in  ac- 
cordance with  the  laws  given  in  §  434,  so  that  they  all  pass 
through  the  point  F. 

The  distance  CF  from  the  center  of  the  lens  to  the  point  at 
which  incident  plane  waves  (parallel  rays)  are  brought  to  a 
focus  is  called  the  focal  length  (/)  of  the  lens. 

The  line  through  the  middle  C  (the  optical  center)  of  the 
lens,  perpendicular  to  its  faces,  is  called  the  principal  axis. 

371 


372 


IMAGE  FORMATION 


FIG.  403.    Focal  plane  of  a  convex  lens 


The  point  F  at  which  rays  parallel  to  the  principal  axis  -are 
brought  to  a  focus  is  called  the  principal  focus. 

The  plane  F'FF"  (Fig.  403)  in  which  plane  waves  (parallel 
rays)  coming  to  the  lens  from  slightly  different  directions,  as 
from  the  top  and  bottom  of  a 
distant  house,  all  have  their 
foci  F',  F",  etc.  is  called  the 
focal  plgne  of  the  lens. 

Since  the  curvature  of  any 
arc  is  denned  as  the  recipro- 
cal of  its  radius  (see  footnote,  p.  364),  the  curvature  which  a  lens 

impresses  on  an  incident  plane  ivave  is  equal  to  -.  Moreover,  no 
matter  what  the  curvature  of  an  incident  wave  may  be,  the 
lens  will  always  change  the  curvature  by  the  same  amount,  —. 

Let  the  focal  length  of  a  convex  lens  be  accurately  determined  by 
measuring  the  distance  from  the  middle  of  the  lens  to  the  image  of  a 
distant  house. 

449.  Conjugate  foci.  If  a  point  source  of  light  is  placed  at 
F  (Fig.  402),  it  is  obvious  that  the  light  which  goes  through 
the  lens  must  exactly  retrace  its  former  path ;  that  is,  its  waves 


FIG.  404.    Conjugate  foci 

will  be  rendered  plane  or  its  rays  parallel  by  the  lens.  But  if 
the  point  source  is  at  a  distance  D0  greater  than  /  ($*£.  404), 

then  the  waves  upon  striking  the  lens  have  a  curvature  — 
(since  the  curvature  of  an  arc  is  denned  as  the  reciprocal  of  its 
radius),  which  is  less  than  their  former  curvature,  -,.  Since  the 
lens  was  able  to  subtract  all  the  curvature  from  waves  coming 


IMAGES  FORMED  BY  LENSES 


373 


from  F  and  render  them  plane,  by  subtracting  the  same 
curvature  from  the  flatter  waves  from  P  it  must  render  them 
concave,;  that  is,  the  rays  after  passing  through  the  lens  are 
converging  and  intersect  at  P'.  If  the  source  is  placed  at  P', 
obviously  the  rays  will  meet  at  P.  The  points  P  and  P1  are 
called  conjugate  foci 

450.  Formula  for  conjugate  foci ;  secondary  foci.  The  rela- 
tion between  the  distances  of  P  and  P1  from  the  lens  is  obtained 
at  once  froni  the  consideration  that  the  curvature  which  the 

h  the  lens,  namely  — ,  is  the  dif- 

-Ls  •  ~t 

impressed  by  the  lens,  namely  - » 
when  it  reached  the  lens,  namely 


wave  has  after  it  gets  t 
ference  between  the  cu 


and  that  which  the  wave 
— ;   that  is, 


/ 


—  = »     or     — - 


(1) 


If  D0  =  D.J  then  the  last  equation  shows  that  both  D0  and  ./>. 
are  equal  to  2/. 

The  two  conjugate  foci  S  and  Sr  which  are  at  equal  dis- 
tances from  the  lens  are  called  the  secondary  foci,  and  their 
distance  from  the  lens  is  twice  the  focal  distance. 

451.  Images  of  Objects.  Let  a  candle  or  electric-light  bulb  be 
placed  between  the  principal  focus  F  and  the  secondary  focus  S  at  PQ 
(Fig.  405),  and  let  a  screen  be  placed  at  P'Q'.  An  enlarged  inverted 
image  will  be  seen  .upon  the  screen. 


FIG.  405.    Formation  of  a  real  image  by  a  lens 

This  image  is  formed  as  follows:  All  the  light  which 
strikes  the  lens  from  the  point  P  is  brought  together  at  a 
point  P'.  The  location  of  this  image  P'  must  be  on  a  straight 


374-  IMAGE  FORMATION 

line  drawn  from  P  through  6';  for  any  ray  passing  through 
C  will  remain  parallel  to  its  original  direction,  since  the  por- 
tions of  the  lens  through  which  it  enters  and  leaves  ma;f  be 
regarded  as  small  parallel  planes  (see  §  434).  The  image 
P'Q'  is  therefore  always  formed  between  the  lines  drawn  from 
P  and  Q  through  C.  If  the  focal  length/  and  the  distance 
of  the  object  D0  are  known,  the  distance  of  the  image  J> 
may  be  obtained  easily  from  formula  (1). 

The  position  of  the  image  may  also  be  found  graphically  as 
follows :  Of  the  c@ne  of  rays  passing  from  P  to  the  lens,  that 


FIG.  406.    Ray  method  of  constructing  image 


ray  which  is  parallel  to  the .  principal  axis  must,  by  §  448, 
pass  through  the  principal  focus  F,'.  The  intersection  of  this 
line  with  the  straight  line  through  C  locates  the  image  P'  (see 
Fig.  406).  $',  the  image  of  Q,  is  located  similarly. 

452.  Size  of  image.  Since  the  image  and  object  are  always 
between  the  intersecting  straight  lines  PP'  and  QQ',  the 
similar  triangles  PCQ  and  P'CQ'  show  that 

'  ~ 


•     Length  of  object  _  Distance  of  object  from  lens 
Length  of  image      Distance  of  image  from  I^ML 

It  may  be  seen  from  Fig.  406,  as  well  as  from  foroulas  (1) 
and  (2),  that 

1.  When  the  object  is  at  S  the  image  is  at  Sr,  and  image 
and  object  are  of  the  same  size. 

2.  As  the  object  moves  out  from  S  to  a  great  distance  the 
image  moves  from  S'  up  to  f,  becoming  smaller  and  smaller. 


IMAGES  FORMED  BY  LENSES 


375 


3.  As  the  object  moves  from  S  up  to  F  the  image  moves  out 
to  a  very  great  distance  to  the  right,  becoming  larger  and  larger. 

4.  When  the  object  is  at  F  the  emerging  waves  are  plane 
(the  emerging  rays  are  parallel),  and  no  rea^l  image  is  formed. 

453.  Virtual  image.  We  have  seen  that  when  the  object  is 
at  F  the  waves  after  passing 
through  the  lens  are  plane.  If, 
then,  the  object  is  nearer  to  the 
lens  than  F,  the  emerging  waves, 
although  reduced  in  curvature  by 


an  amount  - ,  will  still  be  convex, 


FIG.  407.  Virtual  image  formed 

by  a  convex  lens 
and  if  received  by  an  eye  at    #, 

will  appear  to  come  from  a  point  Pr  (Fig.  407).  Since,  however, 
there  is  actually  no  source  of  light  at  Pr,  this  sort  of  image 
is  called  a  virtual  image.  Such 
an  image  cannot  be  projected 
upon  a  screen  as  a  real  image 
can,  but  must  be  observed  by 
an  eye. 

The  graphical  location  of     FJG    4Qg     _Xay  method  of  locating 
a  virtual  image  may  be  ac-  virtual  image  in  convex  lens 

complished    precisely    as    in 

the   case  of  a  real  image   (§451).    It  will  be   seen  that  in 
this  case  (Figs.  407  and  408)  the  image  is  enlarged  and  erect. 

454.  Image  in  concave  lens.    When  a  plane 
wave  strikes  a  concave  lens  it  must  emerge 
as  a  divergent  wave,  since  the  middle  of  the 
wave  is  retarded  by  the  glass  less  than  the 
edges  (Fig.  409).    The  point  F  from  which    FIG.  409.  Virtual 
plane   waves    appear  to    come    after  passing 
through  such  a  lens  is  the  principal  focus  of 
the  lens.    For  the  same  reason  as  in  the  case  of  the  convex 
lens  th  (  centers   of   the  transmitted  waves  from  P  and  Q 


focus   of    a   con- 
cave lens 


376 


IMAGE  FORMATION 


(Fig.  410),  that  is,  the  images  P'  and  Q',  must  lie  upon  the 
lines  PC  and  QC,  and  since  the  curvature  is  increased  by  the 
lens,  they  must  lie  closer  to  the  lens  than  P  and  Q,  Fig.  410 


FIG.  410.    Image  in  a  concave  lens 


FIG.  411.    Ray  method  of  locating 
image  in  concave  lens 


shows  the  way  in  which  such  a  lens  forms  an  image.  This 
image  is  always  virtual,  erect,  and  diminished.  The  graphical 
method  of  locating  the  image  is  shown  in  Fig.  411. 


IMAGES  IN  MIRRORS 

455.  Image  of  a  point  in  a  plane  mirror.  We  are  all  familiar 
with  the  fact  that  to  an  eye  at  E  (Fig.  412),  looking  into  a 
plane  mirror  mti,  a  pencil  point  at  P  appears  to  be  at  some 
point  P'  behind  the  mirror.  We  are  able  in  the  laboratory  to 
find  experimentally  the  exact 
location  of  this  image  P'  with 
respect  to  P  and  the  mirror, 
but  we  may  also  obtain  this 
location  from  theory  as  fol- 
lows :  Consider  a  light  wave 
which  originates  in  the  point 
P  (Fig.  412)  and  spreads  in 
all  directions.  Let  aob  be  a 
section  of  the  wave  at  the 
instant  at  which  it  reaches 
the  reflecting  surface  inn.  An 
instant  later,  if  there  were  no  reflecting  surface,  the  wave 
would  have  reached  the  position  of  the  dotted  line  co^d. 


FIG.   412. 


>* 

Wave   reflected    from    a 
plane  surface 


IMAGES  IN  MIRRORS  377 

Since,  however,  reflection  took  place  at  mn,  and  since  the 
reflected  wave  is  propagated  backward  with  exactly  the  same 
velocity  with  which  the  original  wave  would  have  been  prop- 
agated forward,  at  the  proper  instant,  the  reflected  wave  must 
have  reached  the  position  of  the  line  co^d,  so  drawn  that  oo1 
is  equal  to  oo^.  Now  the  wave  cozd  has  its  center  at  some 
point  P',  and  it  will  be  seen  that  P1  must  lie  just  as  far  below 
mn  as  P  lies  above  it,  for  cofl  and  co^d  are  arcs  of  equal  circles 
having  the  common  chord  cd.  For  the  same  reason,  also,  P1 
must  lie  on  the  perpendicular  drawn  from  P  through  mn. 
When,  then,  a  section  of  this  reflected  wave  co^d  enters  the 
eye  at  E,  the  wave  appears  to  have  originated  at  P'  and  not 
at  P,  for  the  light  actually  comes  to  the  eye  from  P'  as  a 
center  rather  than  from  P.  Hence  P'  is  the  image  of  P. 
We  learn,  therefore,  that  the  image  of  a  point  in  a  plane 
mirror  lies  on  the  perpendicular  drawn  from  the  point  to  the 
mirror,  and  is  as  far  back  of  the  mirror  as  the  point  is  in 
front  of  it. 

456.  Construction  of  image  of  object  in  a  plane  mirror. 
The  image  of  an  object  in  a  plane  mirror  (Fig.  413)  may 
be  located  by  applying  the  law 
proved  above  for  each  of  its 
points,  that  is,  ~by  drawing  from 
each  point  a  perpendicular  to 
the  reflecting  surface,  and  extend- 
ing it  an  equal  distance  on  the 

7        •  7 
other  side. 

To  find  the  path  of  the  rays      Fl(i-  *18:  .  Construction  of  image 

*  of  object  in  plane  mirror 

which  come  to   an  eye  placed 

at  E  from  any  point  such  as  A  of  the  object,  we  have  only  to 
draw  a  line  from  the  image  A1  of  this  point  to  the  eye  and  con- 
nect the  point  of  intersection  of  this  line  with  the  mirror, 
namely  C,  with  the  original*  point  A.  ACE  is  then  the  path 
of  the  fay. 


378 


IMAGE  FORMATION 


Let  a  candle  (Fig.  414)  be  placed  exactly  as  far  in  front  of  a  pane  of 
window  glass  as  a  bottle  full  of  water  is  behind  it,  both  objects  being 
on  the  same  perpendicular  drawn  through 
the  glass.  The  candle  will  appear  to  be 
burning  inside  the  water.  This  explains 
a  large  class  of  familiar  optical  illusions, 
such  as  "the  figure  suspended  in  mid-air," 
the  "  bust  of  a  person  without  a  trunk," 
the  "  stage  ghost,"  etc.  In  the  last  case  the 
illusion  is  produced  by  causing  the  audience 
to  look  at  the  actors  obliquely  through  a 
sheet  of  very  clear  plate  glass,  the  edges  of 


n 


FIG.  414.  Position  of  image 
in  a  plane  mirror 


which  are  concealed  by  draperies.  Images  of  strongly  illuminated  figures 
at  one  side  then  appear  to  the  audience  to  be  in  the  midst  of  the  actors. 

457.  Focal  length  of  a  curved  mirror  half  its  radius  of  curva- 
ture. The  effect  of  a  convex  mirror  on  plane  waves  incident  upon 
it  is  shown  in  Fig.  415.  The  wave  which  would  at  a  given  in- 
stant have  been  at  co^d  is  at  co2d  where  ool  =  oo2.  The  center  F 

from  which  the 

// 
waves  appear  to 

come  to  the  eye 
E  is  the  focus 
of  the  mirror. 
Now  so  long 

as  the  arc  cod  is  A IvAmNl  >  [^ 


4 


\Y 


FIG.  415.    Reflection  of  a  plane  wave  from  a 
convex  mirror 


small  its  curva- 
ture may,  with- 
out appreciable 
error,  be  meas- 
ured by  o^o  (see  footnote,  p.  364) ;  that  is,  by  the  departure 
of  the  curved  line  cod  from  the  straight  line  co^d.  Since  o^ 
was  made  equal  to  ooo,  we  have  o1o2  —  2  o^o  ;  that  is,  the  curva- 
ture -  of  the  reflected  wave  is  equal  to  twice  the  curvature 

R 

of  the  mirror,  or  /  =  — .    In  other  words,  the  focal  length  of  a 

mirror  is  equal  to  one  half  its  radius. 


IMAGES  IN  MIRRORS 


3T9 


Construction  of  image  in  a  con- 
vex mirror 


458.  Image  in  a  convex  mirror.    We  are  all  familiar  with 
the  fact  that  a  convex  mirror  always  forms  behind  the  mirror 
a  virtual,  erect,  and  di- 
minished   image.     The 

reason  for  this  is  shown 
clearly  in  Fig.  416.  The 
image  of  the  point  P 
lies,  as  in  plane  mirrors, 
always  on  the  perpen- 
dicular to  the  mirror,  but     FlG-  416- 
now  necessarily  nearer 
to  the  mirror  than  the  focus  F\  since  the  mirror  has  increased 
the  curvature  of  the  reflected  waves  by  an  amount  equal  to 

— .  The  image  P'Q1  of  an  ob- 
ject PQ  is  always  diminished 
because  it  lies  between  the 
converging  lines  PC  and 
QC.  It  can  be  located  by 
the  ray  method  (Fig.  417)  FIG.  417 

exactly  as  in  the  case  of  con- 
cave lenses.   In  fact,  a  convex  mirror  and  a  concave  lens  have 
exactly  the  same  optical  properties.  This  is  because  each  always 
increases,    the    curvature    of. 
the    incident    waves    l>y    an 

amount  —  • 

459.  Images   in  concave 

mirrors.  Let  the  images  ob- 
tainable with  a  concave  mirror 
be  studied  precisely  as  were 
those  obtainable  from  a  convex 
lens.  It  will  be  found  that  ex- 
actly the  same  series  of  images  is  obtained ;  that  is,  when  the  object  is 
between  the  mirror  and  the  principal  focus  the  image  is  virtual,  enlarged, 


FIG.  418.    Real  image  of  candle  formed 
by  a  concave  mirror 


380 


IMAGE  FOBMATION 


and  erect.  When  it  is  at  the  focus  the  reflected  waves  are  plane;  that  is, 
the  rays  from  each  point  are  a  parallel  bundle.  When  it  is  between  the 
principal  focus  and  the  center  of  curvature,  the  image  is  inverted,  en- 
larged, and  real  (Figs.  418  and  419).  When  it  is  at  a  distance  R  (=  oC) 
from  the  mirror,  the  image  is  also  at  a  distance  R  and  of  the  same  size 


FIG.  419.    Method  of  formation  of  a  real  image  by  a  concave  mirror 

as  the  object,  though  inverted  (see  Fig.  423).  As  the  object  is  moved 
from  R  out  to  a  great  distance  the  im.age^  moves  from  C  up  to  F,  and 
is  always  real,  inverted,  and  diminished.  The  most  convenient  way  of 
finding  the  focal  length  is  to  find  where  the  image  of  a  distant  object 
is  formed. 

We  learn,  then,  that  a  concave  mirror  has  exactly  the  optical 
properties  of  a  convex  lens.    This  is  because,  like  the  convex 


FIG.  420.  Virtual  image  in  a  con- 
cave mirror 


FIG.    421.     Ray   method    of    locating 
real  image  in  concave  mirror- 


lens,  it  always  diminishes  the  curvature  of  the  waves.  The 
same  formulas  hold  throughout,  and  the  same  constructions 
are  applicable  (see  Figs.  420  and  421). 

460.   Summary  for  lenses  and  mirrors.    REAL  IMAGES,  always  Inverted. 
Formed  by  convex  lenses  and  concave  mirrors  if  D0  >f.    Enlarged  if 


1 


I 


T 


J 


5  6 

PHOTOGRAPHS  OF  SOUND  WAVES  HAVING  THEIR  ORIGIN  IN  AN  ELECTRIC 
SPARK  BEHIND  THE  MIDDLE  OF  THE  BLACK  DISK 

1.  A  spherical  sound  wave.  2.  The  same  wave  .00007  second  later.  3.  A  wave  re- 
flected from  a  plane  surface,  curvature  unchanged.  4.  A  Avave  reflected  from  a 
convex  surface,  curvature  increased.  5.  The  source  at  the  focus  of  a  SO2lens.  The 
photograph  shows  flrst,  the  original  wave  on  the  right ;  second,  the  reflected  wave, 
with  its  increased  curvature  ;  and  third,  the  transmitted  plane  wave.  6.  Source  at 
focus  of  a  concave  mirror;  the  reflected  Avave  is  plane.  (Taken  by  Professor  A.L. 
Foley  and  Wilmer  H.  Souder,  of  the  University  of  Indiana) 


IMAGES  IN  MIKROBS  381 

2/  >  1>0  >/;  diminished  if  D0  >  2/.    The  resulting  curvature  is  equal  to 
the  curvature  impressed  by  the  lens,  diminished  by  the  initial  curvature 

A=/~A  A+A  =  /' 

VIRTUAL  IMAGES,  always  erect.  (1)  Formed  by  convex  lenses  and  con- 
cave mirrors  if  A  </>  always  enlarged.  The  resulting  curvature  is 
equal  to  the  initial  curvature  diminished  by  the  curvature  impressed 
by  the  lens.  Ill  111 

A  =  A~/  '     A~A"/' 

(2)  Formed  by  concave  lenses  and  convex  mirrors;  always  dimin- 
ished. The  resulting  curvature  is  equal  to  the  initial  curvature  in- 
creased by  the  curvature  impressed  by  the  lens. 

A^A*/  or  A'AT/' 

The  size  of  all  images  is  given  by 

LO  =  A 

Li      A 

where  L0  and  L{  denote  the  size  of  object  and  image  respectively,  and 
A  and  A  their  distances  from  the  lens  or  mirror.* 

QUESTIONS  AND  PROBLEMS 

1.  A  man  runs  toward  a  plane  mirror  at  the  rate  of  12  ft.  per  second. 
How  fast  does  he  approach  his  image  ? 

2.  A  man  is  standing  squarely  in  front  of  a  plane  mirror  which  is 
very  much  taller  than  he  is.    The  mirror  is  tipped  toward  him  until 
it  makes  an  angle  of  45°  with  the  horizontal.    He  still  sees  his  full 
length.    What  position  does  his  image  occupy? 

3.  Show  from  a  construction  of  the  image  that  a  man  cannot  see 
his  entire  length  in  a  vertical  mirror  unless  the  mirror  is  half  as  tall  as 
he  is.    Decide  from  a  study  of  the  figure  whether  or  not  the  distance  of 
the  man  from  the  mirror  affects  the  case. 

4.  How  tall  is  a  tree  200  ft.  away,  if  the  image  of  it  formed  by. a 
lens  of  focal  length  4  in.  is  1  in.  long  ?    (Consider  the  image  to  be  formed 
in  the  focal  plane.) 

*  Laboratory  experiments  on  the  formation  of  images  by  concave  mirrors  and 
by  lenses  should  follow  this  discussion.  See,  for  example,  Experiments  45  and  46 
of  the  authors'  manual-. 


382 


IMAGE  FORMATION 


5.  How  long  an  image  of  the  same  tree  will  be  formed  in  the  focal 
plane  of  a  lens  having  a  focal  length  of  9  in.  ? 

6.  Why  does  the  nose  appear  relatively  large  in  comparison  with 
the  ears  when  the  face  is  viewed  in  a  convex  mirror? 

7.  Can  a  convex  mirror  ever  form  an  inverted  image  ?    Give  reason 
for  your  answer. 

8.  When  does  a  convex  lens  form  a  real,  and  when  a  virtual,  image  ? 
When  an  enlarged,  and  when  a  diminished,  image?    When  an  erect, 
and  when  an  inverted,  one  ? 

9.  Describe  the  image  formed  by  a  concave  lens.    Why  can  it  never 
be  larger  than  the  object? 

10.  What  is  the  difference' between  a  real  and  a  virtual  image? 

11.  A  candle  placed  20  cm.  in  front  of  a  concave  mirror  has  its  image 
formed  50  cm.  in  front  of  the  mirror.    Find  the  radius  of  the  mirror. 

12.  Find  the  relative  sizes  of  image  and  object  in  Problem  11. 

13.  An  object  is  15  cm. 
in  front  of  a  convex  lens 
of    12   cm.    focal    length. 
What  will  be  the  nature 
of  the  image,  its  size,  and 
its  distance  from  the  lens  ? 

14.  What  is  the  focal 
length  of    a    lens   if    the 
image  of  an  object  10  ft. 
away    is    3  ft.    from    the 
lens? 

15.  If  the  object  in  Problem  14  is  6  in.  long,  how  long  will  the  image  be  ? 

16.  A  beam  of  sunlight  falls  on  a  convex  mirror  through  a  circular 
hole  in  a  sheet  of  cardboard,  as 

in  Fig.  422.  Prove  that  when 
the  diameter  of  the  reflected 
beam  rq  is  twice  the  diameter 
of  the  hole  np,  the  distance  mo 
from  the  mirror  to  the  screen 
is  equal  to  the  focal  length  oF 
of  the  mirror. 

17.  If   a  rose   R   is  pinned 
upside  down  in  a  brightly  illu- 
minated box,  a  real  image  may  be  formed  in  a  glass  of  water  W  by 
a  concave  mirror  C  (Fig.  423).    Where  must  the, eye  be  placed  to  see 
the  image  ? 

18.  How   far  is  the  rose  from  the  mirror  in  the  arrangement  of 
Fig.  423  ? 


FIG.  422.    Determination  of  focal  length  of  a 
convex  mirror 


FIG.  423.    Image  of  object  at  center  of 
curvature 


OPTICAL  INSTRUMENTS 


383 


FIG.  424.    Image   formed   by  a 
small  opening 


OPTICAL  INSTRUMENTS   / 

461.  The  photographic  camera.  A  fairly  distinct,  though 
dim,  image  of  a  candle  can  be  obtained  with  nothing  more 
elaborate  than  a  pinhole  in  a  piece  of  cardboard  (Fig.  424). 
If  the  receiving  screen  is  replaced 
by  a  photographic  plate,  the  ar- 
rangement becomes  a  pinhole 
camera,  with  which  good  pictures 
may  be  taken  if  the  exposure  is 
sufficiently  long.  If  we  try  to 
increase  the  brightness  of  the 
image  by  enlarging  the  hole,  the 
image  becomes  blurred  because  the  narrow  pencils  a,^ ^  a^a'^ 
etc.  become  cones  whose  bases  a\,  a'2,  overlap  and  thus  destroy 
the  distinctness  of  the  outline. 

It  is  possible  to  gain  the  in- 
creased brightness  due  to  the 
larger  hole,  without  sacrificing 
distinctness  of  outline,  by  plac- 
ing a  lens  in  the  hole  (Fig.  425). 
If  the  receiving  screen  is  now 
a  sensitive  plate,  the  arrange- 
ment becomes  a  photographic  camera  (Fig.  426).  But  while, 
with  the  pinhole  camera,  the  screen  may  be  at  any  distance 
from  the  hole,  with  a  lens  the  plate 
and  the  object  must  be  at  conjugate 
foci  of  the  lens. 


Let  a  lens  of,  say,  4  feet  focal  length  be 
placed  in  front  of  a  hole  in  the  shutter  of 
a  darkened  room  and  a  semitransparent 
screen  (for  example,  architect's  tracing 
paper)  placed  at  the  focal  plane.  A  perfect 
reproduction  of  the  opposite  landscape  will 
appear. 


FIG.  425.    Principle  of  the  photo- 
graphic camera 


FIG.  426.  The  photographic 
camera 


384 


IMAGE  FORMATION 


462.  The  projecting  lantern.  The  projecting  lantern  is  essen- 
tially a  camera  in  which  the  position  of  object  and  image  have 
been  interchanged ;  for  in  the  use  of  the  camera  the  object  is 
at  a  considerable  distance  and  a  small  inverted  image  is  formed 
on  a  plate  placed  somewhat  farther  from  the  lens  than  the 
focal  distance.  In  the  use  of  the  projecting  lantern  the  object 
P  (Fig.  427)  is  placed  a  trifle  farther  from  the  lens  L'  than 


FIG.  427.   The  -projecting  lantern  (stereopticon) 


its  focal  length,  and  an  enlarged  inverted  image  is  formed  on 
a  distant  screen  S.  In  both  instruments  the  optical  part  is 
simply  a  convex  lens,  or  a  combination  of  lenses  which  is 
equivalent  to  a  convex  lens. 

The  object  P,  whose  image  is  formed  on  the  screen,  is  usu- 
ally a  transparent  slide  which  is  illuminated  by  a  powerful 
light  A.  The  image  is  as  many  times  larger  than  the  object 
as  the  distance  from  L'  to  S  is  greater  than  the  distance  from 
L'  to  P.  The  light  A  is  usually  either  a  calcium  light  or  an 
electric  arc. 

The  above  are  the  only  essential  parts  of  a  projecting  lantern.  In 
order,  however,  that  the  slide  may  be  illuminated  as  brilliantly  as  pos- 
sible, a  so-called  condensing  lens  L  is  always  used.  This  concentrates 
light  upon  the  transparency  and  directs  it  toward  the  screen. 

In  order  to  illustrate  the  principle  of  the  instrument,  let  a  beam  of 
sunlight  be  reflected  into  the  room  and  fall  upon  a  lantern  slide.  When 
a  lens  is  placed  a  trifle  more  than  its  focal  distance  in  front  of  the  slide, 
a  brilliant  picture  will  be  formed  on  the  opposite  wall. 


OPTICAL  INSTRUMENTS 


385 


463.  The  eye.    The  eye  is  essentially  a  camera  in  which  the 
cornea  C  (Fig.  428),  the  aqueous  humor  Z,  and  the  crystalline 
lens  o  act  as  one  single  lens  which  forms  an  inverted  image 
P'Q'  on  the  retina,  an  expansion  of  the  optic  nerve  covering 
the  inside  of  the  back  of  the  eyeball. 

In  the  case  of  the  camera  the  images  of  objects  at  different 
distances  are  obtained  by  placing  the  plate  nearer  to  or  farther 
from  the  lens.  In  the 
eye,  however,  the  dis- 
tance from  the  retina  to 
the  lens  remains  constant, 
and  the  adjustment  for 
different  distances  is  ef-  FlG.  428.  The  human  eye 

fected  by  changing  the 

focal  length  of  the  lens  itself  in  such  a  way  as  always  to  keep 
the  image  upon  the  retina.  Thus,  when  the  normal  eye  is  per- 
fectly relaxed,  the  lens  has  just  the  proper  curvature  to  focus 
plane  waves  upon  the  retina;  that  is,  to  make  distant  objects 
distinctly  visible.  But  by  directing  attention  upon  near  objects 
we  cause  the  muscles  which  hold  the  lens  in  place  to  contract 
in  such  a  way  as  to  make  the  lens  more  convex,  and  thus 
bring  into  distinct  focus  objects  which  may  be  as  close  as  eight 
or  ten  inches.  This  power  of  adjustment,  however,  varies 
greatly  in  different  individuals. 

464.  The  apparent  size  of  a  body.    The  apparent  size  of  a 
body  depends  simply  upon  the  size  of  the  image  formed  upon 
the  retina  by  the  lens  of  the  eye,     p  p 

and  hence  upon  the  size  of  the 
visual  angle  pCq  (Fig.  429).  TJie 
size  of  this  angle  evidently  in- 
creases as  the  object  is  brought 
nearer  to  the  eye  (see  PCQ). 
Thus  the  image  formed  on  the  retina  when  a  man  is  100  feet 
from  the  eye  is  in  reality  only  one  tenth  as  large  as  the  image 
t 


FIG.  429.   The  visual  angle 


386  IMAGE  FORMATION 

formed  of  the  same  man  when  he  is  but  10  feet  away.  We  do 
not  actually  interpret  the  larger  image  as  representing  a  larger 
man  simply  because  we  have  been  taught  by  lifelong  experi- 
ence to  take  account  of  {he  known  distance  of  an  object  in 
forming  our  estimate  of  its  actual  size.  To  an  infant  who  has 
not  yet  formed  ideas  of  distance  the  man  10  feet  away  doubt- 
less appears  ten  times  as  large  as  the  man  100  feet  away. 

465.  Distance  of  most  distinct  vision.    When  we  wish  to 
examine  an  object  minutely  we  bring  it  as  close  to  the  eye  as 
possible  in  order  to  increase  the  size  of  the  image  on  the  retina. 
But  there  is  a  limit  to  the  size  of  the  image  which  can  be  pro- 
duced in  this  way ;  for  when  the  object  is  brought  nearer  to 
the  normal  eye  than  about  10  inches,  the  curvature  of  the 
incident  wave  becomes  so  great  that  the  eye  lens  is  no  longer 
able,  without  too  much  strain,  to  thicken  sufficiently  to  bring 
the  image  into  sharp  focus  upon  the  retina.    Hence  a  person 
with  normal  eyes  holds  an  object  which  he  wishes  to  see  as 
distinctly  as  possible  at  a  distance  of  about  10  inches. 

Although  this  so-called  distance  of  most  distinct  vision  varies 
somewhat  with  different  people,  for  the  sake  of  having  a 
standard  of  comparison  in  the  determination  of  the  magnifying 
powers  of  optical  instruments,  some  exact  distance  had  to  be 
chosen.  The  distance  so  chosen  is  10  inches,  or  25  centimeters. 

466.  Magnifying  power  of  a  convex  lens.    If  a  convex  lens 
is  placed  immediately  before  the  eye,  the  object  may  be  brought 
much  closer  than  25  centimeters  without  loss  of  distinctness, 
for  the  curvature  of  the  wave  is  partly,  or  even  wholly,  over- 
come by  the  lens  before  the  light  enters  the  eye. 

If  we  wish  to  use  a  lens  as  a  magnifying  glass  to  the  best 
advantage,  we  place  the  eye  as  close  to  it  as  we  can,  so  as  to 
gather  as  large  a  cone  of  rays  as  possible,  and  then  place  the 
object^  at  a  distance  from  the  lens  equal  to  its  focal  length,  so 
that  the  waves  after  passing  through  it  are  plane.  They  are 
then  focused  by  the  eye  with  the  least  possible  effort.  The 


OPTICAL  INSTKUMENTS 


387 


visual  angle  in  such  a  case  is  PcQ  [Fig.  430,  (1)],  for,  since 
the  emergent  waves  are  plane,  the  rays  which  pass  through 
the  center  of  the  eye  from  P  and  Q  are  parallel  to  the  lines 
through  PC  and  Qc.  But  if  the  lens  were  not  present,  and  if 
the  object  were  25  cen- 
timeters from  the  eye, 
the  visual  angle  would 
be  pcq  [Fig.  430,  (2)]. 
The  ratio  of  these  two 
angles  is  approximately 
25//',  where  f  is  the 
focal  length  of  the  lens 
expressed  in  centime- 
ters. Now  the  magnify- 
ing power  of  a  lens  or 
microscope  is  defined  as 

the  ratio  of  the  angle  actually  subtended  by  the  image  when  viewed 
through  the  instrument,  to  the  angle  subtended  by  the  object  when 
viewed  with  the  unaided  eye  at  a  distance  of  25  centimeters. 
Therefore  the  magnifying  power  of  a  simple  lens  is  25/f.  Thus, 
if  a  lens  has  a  focal  length  of  2.5  centimeters,  it  produces  a 
magnification  of  10  diameters  when  the  object  is  placed  at  its 
principal  focus.  If  the  lens  has  a  focal  length  of  1  centimeter, 
its  magnifying  power  is  25,  etc. 

467.  Magnifying  power  of  an  astronomical  telescope.  In  the  astronom- 
ical telescope  the  objective,  or  forward  lens,  forms  at  its  focus  an  image 
of  a  distant  object.  Suppose  that  this  image  were  viewed  directly  by 


FIG.  430.    Magnifying  power  of  a  lens 


FIG.  431.    Magnifying  power  of  a  telescope  objective  is  -F/25 

an  eye  25  centimeters  from  the  image,  as  in  Fig.  431.  The  angle  sub- 
tended by  the  image  at  the  eye  would  then  be  P'EQ' ;  but  the  angle 
subtended  by  the  object  is  PEQ,  which  is  practically  the  same  as  P'cQf ; 


388 


IMAGE  FORMATION 


Magnifying  power  of  a  telescope  is  F/J 


for  P'cQ'  —  PcQ,  and  since  the  object  is  very  distant,  PcQ  =  PEQ 

approximately.    But  P'KQ'  divided  by  P'cQ,'  is  equal  to  F/25,  F  being 

the    focal    length   of   the    objective    measured    in  centimeters.     Hence 

the  forward  lens 

alone  enables  us 

to  increase  the 

visual   angle  of 

the  object  F/25 

times. 

In  practice, 
however,  the  im- 
age is  not  viewed 
with  the  unaided 
eye,  but  with  a 
simple  magnify- 
ing glass  called  an  eyepiece  (Fig.  432),  placed  so  that  the  image  is  at  its 
focus.  Since  we  have  seen  in  §  466  that  the  simple  magnifying  glass 
increases  the  visual  angle  25//  times,  /  being  the  focal  length  of  the 
eyepiece,  it  is  clear  that  the  total  magnification  produced  by  both  lenses, 
used  as  above,  is  F/25  x  25//  =  F/f.  The  magnifying  power  of  an  astro- 
nomical telescope  is  therefore  the  focal  length  of  the  objective  divided  by  the 
focal  length  of  the  eyepiece.  It  will  be  seen, 
therefore,  that  to  get  a  high  magnifying 
power  it  is  necessary  to  use  an  objective 
of  as  great  focal  length  as  possible  and  an 
eyepiece  of  as  short  focal  length  as  possi- 
ble. The  focal  length  of  the  great  lens  at 
the  Yerkes  Observatory  is  about  62  feet 
and  its  diameter  40  inches.  The  great 
diameter  enables  it  to  collect  a  very  large 
amount  of  light. 

Eyepieces  often  have  focal  lengths  as 
small  as  1  inch.  Thus  the  Yerkes  tele- 
scope when  used  with  a  1-inch  eyepiece 
has  a  magnifying  power  of  2976. 

468.  The  magnifying  power  of  the 
compound  microscope.  The  compound 
microscope  is  like  the  telescope  in  that 
the  front  lens,  or  objective,  forms  a  real  image  of  the  object  at  the  focus 
of  the  eyepiece.  The  size  of  the  image  P'Qf  (Fig.  433)  formed  by  the 
objective  is  as  many  times  the  size  of  the  object  PQ  as  v,  the  distance 


eyepiece 


FIG.  433.    The  compound 
microscope 


OPTICAL  INSTRUMENTS 


889 


from  the  objective  to  the  image,  is  times  u,  the  distance  from  the  objec- 
tive to  the  object  (see  §  452).  Since  the  eyepiece  magnifies  this  image 
25//  times,  the  total  magnifying  power  of  a  compound  microscope  is 

—  •   Ordinarily  v  is  practically  the  length  L  of  the  microscope  tube, 

and  u  is  the  focal  length  .F  of  the  objective.    Wherever  this  is  the  case, 

25  L 
then,  the  magnifying  power  of  the  compound  microscope  is  "        • 

The  relation  shows  that  in  order  to  get  a  high  magnifying  power  with 
a  compound  microscope  the  focal  length  of  both  eyepiece  and  objective 
should  be  as  short  as  possible,  while  the  tube  length  should  be  as  long 
as  possible.  Thus,  if  a  microscope  Las  both  an  eyepiece  and  an  objective 
of  6  millimeters  focal  length  and  a  tube  15  centimeters  long,  its  magni- 

25  x  1 5 

fying  power  will  be  —       —  =  1042.    Magnifications  as  high  as  2500  or 
.6  x  .6 

3000  are  sometimes  used,  but  it  is  impossible  to  go  much  farther  for  the 
reason  that  the  image  becomes  too  faint  to  be  seen  when  it  is  spread 
over  so  large  an  area. 

469.  The  terrestrial  telescope.  In  both  the  microscope  and  the  tele- 
scope, since  the  image  formed  by  the  objective  is  a  real  image,  it  is  in- 
verted. Since  the  eyepiece  forms  a  virtual  image  of  this  real  image,  the 
object  as  seen  by  the  eye  will  appear  upside  down.  This  is  a  serious 


FIG.  434.   The  terrestrial  telescope 

objection  when  it  is  desired  to  use  the  telescope  as  a  field  glass.  Hence 
the  terrestrial  telescope  is  constructed  with  an  objective  exactly  like  that 
of  the  astronomical  telescope,  but  with  an  eyepiece  which  is  essentially 
a  compound  microscope.  Since,  then,  the  image  is  twice  inverted,  once 
by  the  objective  0  (Fig.  434)  and  once  by  0',  it  appears  erect. 

470.  The  opera  glass.  On  account  of  the  large  number  of  lenses 
which  must  be  used  in  the  terrestrial  telescope,  it  is  too  bulky  and  awk- 
ward for  many  purposes,  and  hence  it  is  often  replaced  by  the  opera 
glass  (Fig.  435).  This  instrument  consists  of  an. objective  like  that  of 
the  telescope,  and  an  eyepiece  which  is  a  concave  lens  of  the  same  focal 


390 


IMAGE  FORMATION 


length  as  the  eye  of  the  observer.  The  effect  of  the  eyepiece  is  there- 
fore to  just  neutralize  the  lens  of  the  eye.  Hence  the  objective,  in  effect, 
forms  its  image  directly  upon  the  retina.  It  will  be  seen  that  the  size 
of  the  image  formed  upon  the  retina  by  the  objective  of  the  opera  glass 

P'T   -  - 


0 


FIG.  435.    The  opera  glass 

is  as  much  larger  than  the  size  of  the  image  formed  by  the  naked  eye 
as  the  focal  length  CR  of  the  objective  is  greater  than  the  focal  length 
*cR  of  the  eye.  Since  the  focal  length  of  the  eye  is  the  same  as  that  of 
the  eyepiece,  the  magnifying  power  of  the  opera  glass,  like  that  of  the  astro- 
nomical telescope,  is  the  ratio  of  the  focal  lengths  of  the  objective  and  eyepiece. 
Objects  seen  with  an  opera  glass  appear  erect,  since  the  image  formed 
on  the  retina  is  inverted,  as  is  the  case  with  images  formed  by  the  lens 
of  the  eye  unaided. 

471.  The  stereoscope.  Binocular  vision.  When  an  object  is  seen  with 
both  eyes  the  images  formed  on  the  two  retinas  differ  slightly,  because 
of  the  fact  that  the  two  eyes,  on  account  of  their 
lateral  separation,  are  viewing  the  object  from 
slightly  different  angles.  It  is  this  difference  in  the 
two  images  which  gives  to  an  object  or  landscape 
viewed  with  two  eyes  an  appearance  of  depth,  or 
solidity,  which  is  wholly  wanting  when  one  eye  is 
closed.  The  stereoscope  is  an  instrument  which 
reproduces  in  photographs  this  effect  of  binocular 
vision.  Two  photographs  of  the  same  object  are 
taken  from  slightly  different  points  of  view.  These 
photographs  are  mounted  at  A  and  B  (Fig.  436), 
where  they  are  simultaneously  viewed  by  the  two 
eyes  through  the  two  prismatic  lenses  m  and  n. 
These  two  lenses  superpose  the  two  images  at  C 
because  of  their  action  as  prisms,  and  at  the  same  time  magnify  them 
because  of  their  action  as  simple  magnifying  lenses.  The  result  is  that 
the  observer  is  conscious  of  viewing  but  one  photograph  ;  but  this  differs 


FIG.  436.  Principle 
of  the  stereoscope 


OPTICAL  INSTRUMENTS 


391 


from  ordinary  photographs  in  that,  instead  of  being  flat,  it  has  all  of 
the  characteristics  of  an  object  actually  seen  with  both  eyes. 

The  opera  glass  has  the  advantage  over  the  terrestrial  telescope  of 
affording  the  benefit  of  binocular  vision ;  for  while  telescopes  are  usually 
constructed  with  one  tube,  opera  glasses  always  have  two,  one  for  each  eye. 

472.  The  Zeiss  binocular.  The  greatest  disadvantage  of  the  opera 
glass  is  that  the  field  of  view  is  very  small.  The  terrestrial  telescope 
has  a  larger  field,  but  is  of  inconvenient  length.  An  instrument  called  the 
Zeiss  binocular  (Fig.  437)  has 
recently  come  into  use,  which 
combines  the  compactness  of  the 
opera  glass  with  the  wide  field  of 
view  of  the  terrestrial  telescope. 
The  compactness  is  gained  by 
causing  the  light  to  pass  back 
and  forth  through  total  reflect- 
ing prisms,  as  in  the  figure. 
These  reflections  also  perform 
the  function  of  reinverting  the 
image,  so  that  the  real  image 
which  is  formed  at  the  focus  of 
the  eyepiece  is  erect.  It  will  be 
seen,  therefore,  that  the  instrument  is  essentially  an  astronomical  telescope 
in  which  the  image  is  reinverted  by  reflection,  and  in  which  the  tube  is 
shortened  by  letting  the  light  pass  back  and  forth  between  the  prisms. 

A  further  advantage  which  is  gained  by  the  Zeiss  binocular  is  due  to 
the  fact  that  the  two  objectives  are  separated  by  a  distance  which  is 
greater  than  the  distance  between  the  eyes,  so  that  the  stereoscopic 
effect  is  more  prominent  than  with  the  unaided  eye  or  with  the  ordinary 
opera  glass.* 

QUESTIONS  AND   PROBLEMS 

1.  If  a  photographer  wishes  to  obtain  the  full  figure  on  a  plate  of 
cabinet  size,  does  he  place  the  subject  nearer  to  or  farther  from  the 
camera  than  if  he  wishes  to  take  the  head  only?    Why?- 

2.  The  image,  on  the  retina,  of  a  book  held  a  foot  from  the  eye  is 
larger  than  that  of  a  house  on  the  opposite  side  of  the  street.    Why  do 
we  not  judge  that  the  book  is  actually  larger  than  the  house  ? 

*  Laboratory  experiments  on  the  magnifying  powers  of  lenses  and  9n  the  con- 
struction of  microscopes  and  telescopes  should  follow  this  chapter.  See,  for 
example,  Experiments  47,  48,  and  49  of  the  authors'  manual. 


FIG.  437.   The  Zeiss  binocular 


392  IMAGE  FORMATION 

3.  What  is  the  magnifying-  power  of  a  ^-in.  lens  used  as  a  simple 
magnifier  ? 

4.  A  telescope  has  an  objective  of  30  ft.  focal  length  and  an  eyepiece 
of  1  in.  focal  length.  AVhat  is  its  magnifying  power? 

5.  A  stereopticon  is  provided  with  two  lenses,  one  of  6-in.  and  the 
other  of  12-in.  focal  length.    Which  lens  should  be  used  if  it  is  desired 
to  get  as  large  an  image  as  possible  on  a  screen  at  a  fixed  distance  ? 

6.  A  compound  microscope  has  a  tube  length  of  8  in.,  an  objective 
of  focal  length  ?,  in.,  and  an  eyepiece  of  focal  length  1  in.    What  is  its 
magnifying  power  ? 

7.  Explain  why  a  terrestrial  telescope  shows  objects  erect  rather 
than  inverted. 

8.  If  the  focal  length  of  the  eye  is  1  hi.,  what  is  the  magnifying- 
power  of  an  opera  glass  whose  objective  has  a  focal  length  of  4  in.  ? 

9.  If  the  length  of  a  microscope  tube  is  increased  after  an  object 
has  been  brought  into  focus,  must  the  object  be  moved  nearer  to  or 
farther  from  the  lens  in  order  that  the  image  may  be  again  in  focus  ? 

10.  The  magnifying  power  of  a  microscope  is  1000,  the  tube  length 
is  8  in.,  and  the  focal  length  of  the  eyepiece  is  i  in.    What  is  the  focal 
length  of  the  objective  ? 

11.  What  sort  of  lenses  are  necessary  to  correct  shortsightedness? 
longsightedness  ? 


CHAPTER  XX 


COLOR  PHENOMENA 
COLOR  AND  WAVE  LENGTH 
473.  Wave  lengths  of  different  colors.   Let  a  soap  film  be  formed 

across  the  top  of  an  ordinary  drinking  glass,  care  being  taken  that  both 
the  solution  and  the  glass  are  as  clean  as  possible.  Let  a  beam  of  sun- 
light or  the  light  from  a  projecting  lantern  pass  through  a  piece  of 
red  glass  at  A,  fall  upon  the  soap  film  F,  and  be  reflected  from  it  to  a 
white  screen  S  (see  Fig. 438). 
Let  a  convex  lens  L  of  from 
six  to  twelve  inches  focal 
length  be  placed  in  the  path 
of  the  reflected  beam  in  such 
a  position  as  to  produce  an 
image  of  the  film  upon  the 
screen  S,  that  is,  in  such  a 
position  that  film  and  screen 
are  at  conjugate  foci  of  the 
lens.  The  system  of  red  and 
black  bands  upon  the  screen 
is  formed  precisely  as  in 
§  438,  by  the  interference  of 
the  two  beams  of  light  com- 
ing from  the  front  and  back 
surf  aces  of  the  wedge-shaped  FIG.  438.  Projection  o:f  soap-film  fringes 
film.  Let  now  the  red  glass 

be  held  in  one  half  of  the  beam  and  a  piece  of  green  glass  in  the  other 
half,  the  two  pieces  being  placed  edge  to  edge,  as  shown  at  A.  Two 
sets  of  fringes  will  be  seen  side  by  side  on  the  screen.  The  fringes  will 
be  red  and  black  on  one  side  of  the  image  and  green  and  black  on 
the  other;  but  it  will  be  noticed  at  once  that  the  dark  bands  on  the 
green  side  are  closer  together  than  the  dark  bands  on  the  other  side; 


394  COLOR  PHENOMENA 

in  fact,  seven  fringes  on  the  side  of  the  film  which  is  covered  by  the 
green  glass  will  be  seen  to  cover  about  the  same  distance  as  six  fringes 
on  the  red  side.* 

Since  it  was  shown  in  §  440  that  the  distance  between  two 
dark  bands  corresponds  to  an  increase  of  one-half  wave  length 
in  the  thickness  of  the  film,  we  conclude,  from  the  fact  that 
the  dark  bands  on  the  red  side  are  farther  apart  than  those  on 
the  green  side,  that  red  light  must  have  a  longer  wave  length 
than  green  light.  The  wave  length  of  the  central  portion  of 
each  colored  region  of  the  spectrum  is  roughly  as  follows : 

Red 000068  cm.         Yellow 000058  cm. 

Green 000052  cm.         Blue 000046  cm. 

Violet 000042  cm. 

Let  the  red  and  green  glasses  be  removed  from  the.path  of  the  beam. 
The  red  and  green  fringes  will  be  seen  to  be  replaced  by  a  series  of 
bands  brilliantly  colored  in  different  hues.  These  are  due  to  the  fact 
that  the  lights  of  different  wave  length 
form  interference  bands  at  different 
places  on  the  screen.  Notice  that  the 
upper  edges  of  the  bands  (lower  edges 
in  the  inverted  image)  are  reddish, 
while  the  lower  edges  are  bluish.  We 
shall  find  the  explanation  of  this  fact 
in  §  482. 

474.  Composite  nature  of  white 

light.    Let  a  beam  of  sunlight  pass 

through  a  narrow  slit  and  fall  on  a       FIG.  439.    White  light  decom- 

prism,  as  in  Fig.  439.   The  beam  which  posed  by  a  prism 

enters    the    prism    as    white    light   is 

broken  up  into  red,  yellow,  green,  blue,  and  violet  lights,  although  each 

color  merges,  by  insensible  gradations,  into  the  next.    This  band  of 

color  is  called  a  spectrum. 

We  conclude  from  this  experiment  that  white  light  is  a  mix- 
ture of  all  the  colors  of  the  spectrum,  from  red  to  violet  inclusive. 

*  The  experiment  may  be  performed  at  home  by  simply  looking  through  red 
and  green  glasses  at  a  soap  film  so  placed  as  to  reflect  white  light  to  the  eye. 


COLOK  AND  WAVE  LENGTH  395 

475.  Color  Of  bodies  in  white  light.    Let  a  piece  of  red  glass  be 
held  in  the  path  of  the  colored  beam  of  light  in  the  experiment  of  the 
preceding  section.    All  the  spectrum  except  the  red  will  disappear,  thus 
showing  that  all  the  wave  lengths  except  red  have  been  absorbed  by  the 
glass.    Let  a  green  glass  be  inserted  in  the  same  way.    The  green  portion 
of  the  spectrum  will  remain  strong,  while  the  other  portions  will  be 
greatly  enfeebled.    Hence  green  glass  must  have  a  much  less  absorbing 
effect  upon  wave  lengths  which  correspond  to  green  than  upon  those 
which  correspond  to  red  and  blue.    Let  the  green  and  red  glasses  be  held 
one  behind  the  other  in  the  path  of  the  beam.    The  spectrum  will  almost 
completely  vanish,  for  the  red  glass  has  absorbed  all  except  the  red  rays, 
and  the  green  glass  has  absorbed  these. 

We  conclude,  therefore,  that  the  color  which  a  body  has  in 
ordinary  daylight  is  determined  by  the  wave  lengths  which 
the  body  has  not  the  power  of  absorbing.  Thus,  if  a  body 
appears  white  in  daylight,  it  is  because  it  diffuses  or  reflects 
all  wave  lengths  equally  to  the  eye,  and  does  not  absorb  one 
set  more  than  another.  For  this  reason  the  light  which  comes 
from  it  to  the  eye  is  of  the  same  composition  as  daylight  or 
sunlight.  If,  however,  a  body  appears  red  in  daylight,  it  is 
because  it  absorbs  the  red  rays  of  the  white  light  which  falls 
upon  it  less  than  it  absorbs  the  others,  so  that  the  light  which 
is  diffusely  reflected  contains  a  larger  proportion  of  red  wave 
lengths  than  is  contained  in  ordinary  light.  Similarly,  a  body 
appears  yellow,  green,  or  blue  when  it  absorbs  less  of  one  of 
these  colors  than  of  the  rest  of  the  colors  contained  in  white 
light,  and  therefore  sends  a  preponderance  of  some  particular 
wave  length  to  the  eye. 

476.  Color  of  bodies  placed  in  colored  lights.  Let  a  body  which 

appears  white  in  sunlight  be  placed  in  the  red  end  of  the  spectrum.  It 
will  appear  to  be  red.  In  the  blue  end  of  the  spectrum  it  will  appear  to 
be  blue,  etc.  This  confirms  the  conclusion  of  the  last  paragraph,  that 
a  white  body  has  the  power  of  diffusely  reflecting  all  the  colors  of  the 
spectrum  equally. 

Next  let  a  skein  of  red  yarn  be  held  in  the  blue  end  of  the  spec- 
trum. It  will  appear  nearly  black.  In  the  red  end  of  the  spectrum 


396 


COLOR  PHENOMENA 


it  will  appear  strongly  red.  Similarly,  a  skein  of  blue  yarn  will  appear 
nearly  black  in  all  the  colors  of  the  spectrum  except  blue,  where  it 
will  have  its  proper  color. 

These  effects  are  evidently  due  to  the  fact  that  the  red  yarn, 
for  example,  has  the  power  of  diffusely  reflecting  red  wave 
lengths  copiously,  but  of  absorbing,  to  a  large  extent,  the  others. 
Hence,  when  held  in  the  blue  end  of  the  spectrum,  it  sends 
but  little  color  to  the  eye,  since  no  red  light  is  falling  upon  it. 

477.  Compound  colors.  It  must  not  be  inferred  from  the 
preceding  paragraphs  that  every  color  except  white  has  one 
definite  wave  length,  for  the  same  effect 
may  be  produced  on  the  eye  by  a  mix- 
ture of  several  different  wave  lengths  as 
is  produced  by  a  single  wave  length. 
This  statement  may  be  proved  by  the 
use  of  an  apparatus  known  as  Newton's 
color  disk  (Fig.  440).  The  arrangement 
makes  it  possible  to  rotate  differently 
colored  sectors  so  rapidly  before  the  eye 
that  the  effect  is  precisely  the  same  as 
though  the  colors  came  to  the  eye  simul- 
taneously. If  one  half  of  the  disk  is  red 
and  the  other  half  green,  the  rotating 
disk  will  appear  yellow,  the  color  being 
very  similar  to  the  yellow  of  the  spec- 
trum. If  green  and  violet  are  mixed  in 
the  same  way,  the  result  will  be  light  blue.  Although  the 
colors  produced  in  this  way  are  not  distinguishable  by  the  eye 
from  spectral  colors,  it  is  obvious  that  their  physical  constitu- 
tion is  wholly  different ;  for  while  a  spectral  color  consists  of 
waves  of  a  single  wave  length,  the  colors  produced  by  mix- 
ture are  compounds  of  several  wave  lengths.  For  this  reason 
the  spectral  colors  are  called  pure  and  the  others  compound. 
In  order  to  tell  whether  the  color  of  an  object  is  pure  or 


FIG.  440.   Newton's 
color  disk 


COLOR  AND  WAVE  LENGTPI  397 

compound,  it  is  only  necessary  to  observe  it  through  a  prism. 
If  it  is  compound,  the  colors  will  be  separated,  giving  an  image 
of  the  object  for  each  color.  If  it  is  pure,  the  object  will  appear 
through  the  prism  exactly  as  it  does  without  the  prism. 

By  compounding  colors  in  the  way  described  above  we 
can  produce  many  which  are  not  found  in  the  spectrum. 
Thus  mixtures  of  red  and  blue  give  purple  or  crimson ;  mix- 
tures of  black  with  red,  orange,  or  yellow  give  rise  to  the 
various  shades  of  brown.  Lavender  may  be  formed  by  add- 
ing three  parts  of  white  to  one  of  blue ;  lilac,  by  adding  to 
Fifteen  parts  of  white  four  parts  of  red  and  one  of  blue ; 
olive,  by  adding  one  part  of  black  to  two  parts  of  green  and 
one  of  red. 

478.  Complementary  colors.  Since  white  light  is  a  combina- 
tion of  all  the  colors  from  red  to  violet  inclusive,  it  might  be 
expected  that  if  one  or  several  of  these  colors  were  subtracted 
from  white  light,  the  residue  would  be  colored  light. 

To  test  this  experimentally  let  a  beam  of  sunlight  be  passed  through 
a  slit  s,  a  prism  P,  and  a  lens  L,  to  a  screen  S,  arranged  as  in  Fig.  441. 
A  spectrum  will  be  formed  at  R  V,  the  position  conjugate  to  the  slit  s, 
and  a  pure  white  spot  will  ap- 
pear on  the  screen  when  it  is 
at  the  position  which  is  con- 
jugate to  the  prism  face  ab. 
Let  a  card  be  slipped  into 
the  path  of  the  beam  at  R, 
so  as  to  cut  off  the  red  portion 
of  the  light.  The  spot  on  S 
will  appear  a  brilliant  shade  FR,  441  Hecombinatioll  of  spectral  colors 
of  greenish  blue.  This  is  the  into  white  ijght 

compound  color  left  after  red 

is  taken  from  the  white  light.  This  shade  of  blue  is  therefore  called  the 
complementary  color  of  the  red  which  has  been  subtracted.  Two  comple- 
mentary colors  are  therefore  denned  as  any  two  colors  which  produce 
white  when  added  to  each  other. 

Let  the  card  be  slipped  in  from  the  side  of  the  blue  rays  at  V.  The 
spot  will  first  take  on  a  yellowish  tint  when  the  violet  alone  is  cut  out ; 


398  COLOR  PHENOMENA 

and  as  the  card  is  slipped  farther  in,  the  image  will  become  a  deep  shade 
of  red  when  violet,  blue,  and  part  of  the  green  are  cut  out. 

Next  let  a  lead  pencil  be  held  vertically  between  R  and  Fso  as  to  cut  off 
the  middle  part  of  the  spectrum;  that  is,  the  yellow  and  green  rays.  The  re- 
maining red,  blue,  and  violet  will  unite  to  form  a  brilliant  purple.  In  .each 
case  the  color  on  the  screen  is  the  complement  of  that  which  is  cut  out. 

479.  Retinal  fatigue.    Let  the  gaze  be  fixed  intently  for  not  less 
than  twenty  or  thirty  seconds  on  a  point  at  the  center  of  a  block  of  any 
brilliant  color  —  for  example,  red.    Then  look  off  at  a  dot  on  a  white 
wall  or  a  piece  of  white  paper,  and  hold  the  gaze  fixed  there  for  a  few 
seconds.    The  brilliantly  colored  block  will  appear  on  the  white  wall, 
but  its  color  will  be  the  complement  of  that  first  looked  at. 

The  explanation  of  this  phenomenon,  due  to  so-called  "ret- 
inal fatigue,"  is  found  in  the  fact  that  although  the  white  sur- 
face is  sending  waves  of  all  colors  to  the  eye,  the  nerves  which 
responded  to  the  color  first  looked  at  have  become  fatigued, 
and  hence  fail  to  respond  to  this  color  when  it  comes  from  the 
white  surface.  Therefore  the  sensation  produced  is  that  due 
to  white  light  minus  this  color ;  that  is,  to  the  complement  of 
the  original  color.  A  study  of  the  spectral  colors  by  this 
method  shows  that  the  following  colors  are  complementary. 

Red  Orange  Yellow         Violet  Green 

Bluish  green         Greenish  blue         Blue  Greenish  yellow        Crimson 

480.  Color  of  pigments.    When  yellow  light  is  added  to  the 
proper  shade  of  blue,  white  light  is  produced,  since  these 
colors  are  complementary.    But  if  a  yellow  pigment  is  added 
to  a  blue  one,  the  color  of  the  mixture  will  be  green.    This  is 
because  the  yellow  pigment  removes  the  blue  and  violet  by 
absorption,  and  the  blue  pigment  removes  the  red  and  yellow, 
so  that  only  green  is  left. 

When  pigments  are  mixed,  therefore,  each  one  subtracts  cer- 
tain colors  from  white  light,  and  the  color  of  the  mixture  is  that 
color  which  escapes  absorption  by  the  different  ingredients. 
Adding  pigments  and  adding  colors,  as  in  §  477,  are  therefore 
wholly  dissimilar  processes  and  produce  widely  different  results. 


COLOR  AND  WAVE  LENGTH  399 

481.  Three-color  printing.    It  is  found  that  all  colors  can  be 
produced  by  suitably  mixing  with  the  color  disk  (Fig.  440) 
three    spectral    colors,   namely   red,    green,    and   blue-violet. 
These  are  therefore  called  the  three  primary  colors.    The  so- 
called  primary  pigments  are  simply  the  complements  of  these 
three  primary  colors.    They  are,  in  order,  peacock  blue,  crim- 
son, and  light  yellow.    The  three  primary  colors  when  mixed 
yield  white.    The  three  primary  pigments  when  mixed  yield 
black,  because  together  they  subtract  all  the  ingredients  from 
white  light.    The  process  of  three-color  printing  consists  in 
mixing  on  a  white  background,  that  is,  on  white  paper,  the 
three  primary  pigments  in  the  following  way :  Three  differ- 
ent photographs  of  a  given-colored  object  are  taken,  each 
through  a  filter  of  gelatin  stained  the  color  of  one  of  the 
primary  colors.    From  these  photographs  half-tone  "  blocks  " 
are  made  in  the  usual  way.   The  colored  picture  is  then  made 
by  carefully  superposing  prints  from  these  blocks,  using  with 
each  an  ink  whose  color  is  the  complement  of  that  of  the 
"  filter  "  through  which  the  original  negative  was  taken.   The 
plate  opposite  page  400  illustrates  fully  the  process.    It  will 
be  interesting  to  examine  differently  colored  portions  with  a 
lens  of  moderate  magnifying  power. 

482.  Colors  of  thin  films.   The  study  of  complementary  colors 
has  furnished  us  with  the  key  to  the  explanation  of  the  fact, 
observed  in  §  473,  that  the  upper  edge  of  each  colored  band 
produced  by  the  water  wedge  is  reddish,  while  the  lower  edge 
is  bluish.    The  red  on  the  upper  edge  is  due  to  the  fact  that 
there  the  shorter  blue  waves  are  destroyed  by  interference  and 
a  complementary  red  color  is  left ;  while  on  the  lower  edge 
of  each  fringe,  .where  the  film  is  thicker,  the  longer  red  waves 
interfere,  leaving  a  complementary  blue.    In  fact,  each  wave 
length  of  the  incident  light  produces  a  set  of  fringes,  and  it  is 
the  superposition  of  these  different  sets  which  gives  the  result- 
ant colored  fringes.   Where  the  film  is  too  thick  the  overlapping 


400  COLOR  PHENOMENA 

is  so  complete  that  the  eye  is  unable  to  detect  any  trace  of 
color  in  the  reflected  light. 

In  films  which  are  of  uniform  thickness,  instead  of  wedge- 
shaped,  the  color  is  also  uniform,  so  long  as  the  observer  does 
not  change  the  angle  at  which  the  film  is  viewed.  With  any 
change  in  this  angle  the  thickness  of  film  through  which  the 
light  must  pass  in  coming  to  the  observer  changes  also,  and 
hence  the  color  changes.  This  explains  the  beautiful  play  of 
iridescent  colors  seen  in  soap  bubbles,  thin  oil  films,  mother 
of  pearl,  etc. 

483.  Chromatic  aberration.    It  has  heretofore  been  assumed 
that  all  the  waves  which  fall  on  a  lens  from  a  given  source 
are  brought  to  one  and  the  same  focus.    But  since  blue  rays 
are  bent  more  than  red  ones  in  passing  through  a  prism,  it  is 
clear  that  in  passing  through  a  lens  the  blue  light  must  be 
brought  to  a  focus  at  some  point  v  (Fig.  442)  nearer  to  the 
lens  than  r,  where  the  red  light  is  focused,  and  that  the  foci  for 
intermediate  colors  must  fall  in  intermediate  positions.    It  is 
for  this  reason  that  an  image 

formed  by  a  simple  lens  is 

always  fringed  with  color. 

V  v    r 

Let  a  card  be  held  at  the  pIG.  442.  Chromatic  aberration  in  a  lens 
focus  of  a  lens  placed  in  a  beam 

of  sunlight  (Fig.  442).  If  the  card  is  slightly  nearer  the  lens  than  the 
focus,  the  spot  of  light  will  be  surrounded  by  a  red  fringe,  for  the  red 
rays,  being  least  refracted,  are  on  the  outsi^fi.  If  the  card  is  moved  out 
beyond  the  focus,  the  red  fringe  will  be  found  to  be  replaced  by  a  blue 
one  ;  for,  after  crossing  at  the  focus,  it  will  be  the  more  refrangible  rays 
which  will  then  be  found  outside. 

This  dispersion  of  light  produced  by  a  lens  is  called  chromatic 
aberration. 

484.  Achromatic  lenses.    The   color  effect  caused  by  the 
chromatic  aberration  of  a  simple  lens  greatly  impairs  its  use- 
fulness.   Fortunately,  however,  it  has  been  found  possible  to 


THREE-COLOR  PRINTING 

1.  Yellow  impression;  negative  made  through  a  blue-violet  filter.  2.  Crim- 
son impression ;  negative  made  through  a  green  filter.  3.  Crimson  on  yellow. 
4.  Blue  impression  ;  negative  made  through  a  red  filter.  5.  Yellow,  crimson,  and 
blue  combined  ;  the  final  product. 

The  squares  at  the  left  show  the  colors  of  ink  used  in  making  each  impression. 
Notice  the  different  colors  in  5,  which  are  made  by  combining  the  yellow,  crimson, 
and  blue. 


SPECTRA  101 

eliminate  this  effect  almost  completely  by  combining  in(.n 
one  lens  a  convex  Ions  of  crown  glaHH  and  a  concave  lens 
of  Hint  glass  (Fig.  4-ltt).  The  first  Ions  Oww 

then    produces    both   bending    and    disper- 
sion, while   the  second   almost  completely        IS 

«/  ^\\ 

overcomes  the  dispersion  without  entirely 

overcoming  the  bending.    Sueh   lenses  are     l''i<».  44JJ.  Anaohro- 

,.,,          ,.  UWlle  leilS 

called  flkTArotnofifl  fcnw*.    1  he  lirst  ono  was 

made  by  John   Dollond   in   London   in    1758.    They  are  used 

in   (he  construction   of  all  good   telescopes  and   microscopes. 

QUESTIONS  AND  PROBLEMS 

1.  If  a  soap  lilin  is  illuminated  with  red,  ^HHMI,  and  yollow   n  M>    of 
li^hl.,  side  l>y  side,  how   will  the  distance  between  the  yellow   fringes 
com  pa  iv  with  that  between  the  red  fringe!  ¥  with  that  between  the  g'reen 
fringes?    (See  table  on  pag'e  JMM.) 

2.  \\  hat  will  I  "•  Hi--  .1 1  >|  MI  '-MI  .-..I,. i  of  a  red  body  when  it  is  in  a  room 
to  which  only  ^rcen  lijjht  is  Admitted? 

3.  Why  do  whiti^  bodi(«s  look  liluc  when  Heen  thnmgh  a  blue  glass? 

4.  What  color  would  a  yellow  object  appear  to  have  if  looked  at 
through  a  blue  glass?    (Assume  that  the  yellow  is  a  pure  color.) 

5.  A  gas  llatne  is  distinctly  yellow  as  compared  with  sunlight.    What 
wave  lengths,  then,  inn  .1   he  comparatively  weak  in  the  spectrum  of  a 
gas  tlame? 

6.  If  the  green  and  the  yellow  are  subtracted  from  white  light,  what 
will  be  the  color  of  the  residue? 

7.  Will  a  reddish  spot  on  an  oil  (Urn  be  thinner  or  thicker  than  an 
adjacent  bluish  portion? 

SPECTRA 

485.  The  rainbow.  A  very  beautiful  spectrum  with  which 
every  one  is  familiar  is  the  rotftfotft 

Lot  a  Hpherical  bulb  /''(1'Mg.  4-M)  1  i  or  fJ  Inohcn  In  diameter  be  filled 
with  water  and  held  in  the  path  of  a  beam  of  Nimlight  which  (Miters  I  he 
room  through  a  hole  in  a  pieoe  of  cardboard  All.  A  miniature  rainl»>" 
will  be  formed  on  the  Noroen  around  the  opening,  the  violet  edge  of  UK 
bow  being  toward  the  center  of  the  circle  and  the  red  outside.  A  beam  »i 
light,  which  .-ni.-i  •  Hi.-  (task  at  '  is  there  both  refracted  and  dispersed  ; 

I 


402 


COLOR  PHENOMENA 


FIG.  444.    Artificial  rainbow 


at  D  it  is  totally  reflected ;  and  at  E  it  is  again  refracted  and  dispersed 
on  passing  out  into  the  air.  Since  in  both  of  the  refractions  the  violet  is 
bent  more  than  the  red,  it  is  obvious  that  it  must  return  nearer  to  the 
direction  of  the  incident 
beam  than  the  red  rays. 
If  the  flask  were  a  perfect 
sphere,  the  angle  included 
between  the  incident  ray 
OC  and  the  emergent  red 
ray  ER  would  be  42° ;  and 
the  angle  between  the  inci- 
dent ray  and  the  emergent 
violet  ray  E  Fwould  be  40°. 

The  actual  rainbow 
seen  in  the  heavens  is 
due  to  the  refraction 
and  reflection  of  light  in  the  drops  of  water  in  the  air,  which 
act  exactly  as  did  the  flask  in  the  preceding  experiment.  If  the 
observer  is  standing  at  E  with  his  back  to  the  sun,  the  light 
which  comes  from  the  drops  so  as  to  make  an  angle  of  42° 
(Fig.  445)  with  the  line  drawn  from  the  observer  to  the  sun 
must  be  red  light ;  while  the  light  which  comes  from  drops 
which  are  at  an  angle 
of  40°  from  the  eye 
must  be  violet  light. 
It  is  clear  that  those 
drops  whose  direction 
from  the  eye  makes 
any  particular  angle 

with  the  line   drawn 

,.  FIG.  445.    Primary  and  secondary  rainbows 

irom  the  eye  to   the 

sun  must  lie  on  a  circle  whose  center  is  on  that  line.  Hence 
we  see  a  circular  arc  of  light  which  is  violet  on  the  inner  edge 
and  red  on  the  outer  edge. 

486.  The  secondary  bow.    A  second  bow  having  the  red  on 
the  inside  and  the  violet  on  the  outside  is  often  seen  outside 


SPECTRA 


403 


of  the  one  just  described,  and  concentric  with  it.  This  bow 
arises  from  rays  which  have  suffered  two  internal  reflections 
and  two  refractions,  in  the  manner  shown  in  Fig.  445.  Since 
in  this  bow  the  emergent  ray  crosses  the  incident  ray,  it  will 
be  seen  that  the  color  which  suffers  the  largest  refraction  must 
make  the  largest  angle  with  the  incident  ray.  Hence  the  violet 
which  comes  to  the  eye  must  come  from  drops  which  are  farther 
from  the  center  of  the  circle  than  those  which  send  the  red 
The  red  rays  come  from  an  angle  of  51°  and  the  violet  rays 
from  an  angle  of  54°. 

487.  Continuous  spectra.   Let  a  Bunsen  burner  arranged  to  produce 

a  white  flame  be  placed  behind  a  slit  s  (Fig.  446).  Let  the  slit  be  viewed 
through  a  prism.  P.  The  spectrum  will  be  a  continuous  band  of  color. 
If,  then,  the  air  is  admitted  at 
the  base  of  the  burner,  and 
if  a  clean  platinum  wire  is 
held  in  the  flame  directly  in 
front  of  the  slit,  the  white-hot 
platinum  will  also  give  a  con- 
tinuous spectrum.* 

All  incandescent  solids 
and  liquids  are  found  to 
give  spectra  of  this  type 
which  contain  all  the  wave  lengths  from  the  extreme  red  to 
the  extreme  violet.  The  continuous  spectrum  of  a  luminous  gas 
flame  is  due  to  the  incandescence  of  solid  particles  of  carbon 
suspended  in  the  flame.  The  presence  of  these  solid  particles 
is  proved  by  the  fact  that  soot  is  deposited  on  bodies  held 
in  a  white  flame. 

488.  Bright-line  Spectra.    Let  a  bit  of  asbestos  or  a  platinum  wire 
be  dipped  into  a  solution  of  common  salt  (sodium  chloride)  and  held  in 
the  flame,  care  being  taken  that  the  wire  itself  is. held  so  low  that  the 

*  By  far  the  most  satisfactory  way  of  performing  these  experiments  with  spectra 
is  to  provide  the  class  with  cheap  plate-glass  prisms  like  those  used  in  Experi- 
ment 50  of  the  authors'  manual,  rather  than  to  attempt  to  project  line  spectra. 


FIG.  446.  Arrangement  for  viewing  spectra 


404  COLOR  PHENOMENA 

spectrum  due  to  it  cannot  be  seen.  The  continuous  spectrum  of  the 
preceding  paragraph  will  be  replaced  by  a  clearly  denned  yellow  image 
of  the  slit  which  occupies  the  position  of  the  yellow  portion  of  the 
spectrum.  This  shows  that  the  light  from  the  sodium  flame  is  not  a 
compound  of  a  number  of  wave  lengths,  but  is  rather  of  just  the  wave 
length  which  corresponds  to  this  particular  shade  of  yellow.  The  light 
is  now  coming  from  the  incandescent  sodium  vapor  and  not  from  an 
incandescent  solid,  as  in  the  preceding  experiments. 

Let  another  platinum  wire  be  dipped  in  a  solution  of  lithium  chloride 
and  held  in  the  flame.  Two  distinct  images  of  the  slit,  s'  and  s" 
(Fig.  446),  will  be  seen,  one  in  yellow  and  one  in  red.  Let  calcium 
chloride  be  introduced  into  the  flame.  One  distinct  image  of  the  slit 
will  be -seen  in  the  green  and  another  in  the  red.  Strontium  chloride 
will  give  a  blue  and  a  red  image,  etc.  (The  yellow  sodium  image  will 
probably  be  present  in  each  case,  because  sodium  is  present  as  an 
impurity  in  nearly  all  salts.) 

These  narrow  images  of  the  slit  in  the  different  colors  are 
called  the  characteristic  spectral  lines  of  the  substances.  The 
experiments  show  that  incandescent  vapors  and  gases  give  rise 
to  bright-line  spectra,  and  not  continuous  spectra  like  those 
produced  by  incandescent  solids  and  liquids.  The  method  of 
analyzing  compound  substances  through  a  study  of  the  lines 
in  the  spectra  of  their  vapors  is  called  spectrum  analysis.  It 
was  first  used  by  the  German  chemist  Bunsen  in  1859. 

489.  The  solar  Spectrum.  Let  a  beam  of  sunlight  pass  first 
through  a  narrow  slit  S  (Fig.  447),  not  more  than  £  millimeter  in  width, 
then  through  a  prism  P,  and  finally  let  it  fall  on  a  screen  £',  as  shown  in 
Fig.  447.  Let  the  position  of  the  prism  be  changed  until  a  beam  of 
white  light  is  reflected  from  one  of  its  faces  to  that  portion  of  the  screen 
which  was  previously  occupied  by  the  central  portion  of  the  spectrum. 
Then  let  a  lens  L  be  placed  between  the  prism  and  the  slit,  and  moved 
back  and  forth  until  a  perfectly  sharp  white  image  of  the  slit  is  formed 
on  the  screen.  This  adjustment  is  made  in  order  to  get  the  slit  S  and 
the  screen  S'  in  the  positions  of  conjugate  foci  of  the  lens.  Now  let  the 
prism  be  turned  to  its  original  position.  The  spectrum  on  the  screen 
will  then  consist  of  a  series  of  colored  images  of  the  slit  arranged  side 
by  side.  This  is  called  a  pure  spectrum,  to  distinguish  it  from  the  spec- 
trum shown  in  Fig.  439,  in  which  no  lens  was  used  to  bring  the  rays  of 


SPECTRA  405 

each  particular  color  to  a  particular  point,  and  in  which  there  was 
therefore  much  overlapping  of  the  different  colors.  If  the  slit  and  screen 
are  exactly  at  conjugate  foci  of  the  lens,  and  if  the  slit  is  sufficiently 
narrow,  the  spectrum  will  be 
seen  to  be  crossed  vertically  by 
certain  dark  lines. 

These  lines  were  first  ob- 
served  by  the  Englishman 
Wollaston  in  1802,  and 
were  first  studied  carefully 

by  the  German  Fraunhofer 

..      FIG.  447.    Arrangement  for  obtaining  a 

in  1814,  who  counted  and  pure  spectrum 

mapped    out    as    many    as 

seven  hundred  of  them.  They  are  called  after  him,  the  Fraun- 
hofer lines.  Their  existence  in  the  solar  spectrum  shows  that 
certain  wave  lengths  are  absent  from  sunlight,  or,  if  not  entirely 
absent,  are  at  least  much  weaker  -than  their  neighbors.  When 
the  experiment  is  performed  as  described  above  it  will  usually 
not  be  possible  to  count  more  than  five  or  six  distinct  lines. 

490.  Explanation  of  the  Fraunhofer  lines.  Let  the  solar  spec- 
trum be  projected  as  in  §  489.  Let  a  few  small  bits  of  metallic  sodium 
be  laid  upon  a  loose  wad  of  asbestos  which  hns  been  saturated  with 
alcohol.  Let  the  asbestos  so  prepared  be  held  to  the  left  of  the  slit,  or 
between  the  slit  and  the  lens,  and  there  ignited.  A  black  band  will  at 
once  appear  in  the  yellow  portion  of  the  spectrum,  at  the  place  where 
the  color  is  exactly  that  of  the  sodium  flame  itself ;  or  if  the  focus  was 
sufficiently  sharp  so  that  a  dark  line  could  be  seen  in  the  yellow  before 
the  sodium  was  introduced,  this  line  will  grow  very  much  blacker  when 
the  sodium  is  burned.  Evidently  then  this  dark  line  in  the  yellow  part 
of  the  solar  spectrum  is  due  in  some  way  to  sodium  vapor  through  which 
the  sunlight  has  somewhere  passed. 

The  experiment  at  once  suggests  the  explanation  of  the 
Fraunhofer  lines.  The  white  light  which  is  emitted  by  the 
hot  nucleus  of  the  sun,  and  which  contained  all  wave  lengths, 
has  had  certain  wave  lengths  weakened  by  absorption  as  it 


406 


COLOK  PHENOMENA 


passed  through  the  vapors  and  gases  surrounding  the  sun  and 
the  earth.  For  it  is  found  that  every  gas  or 
vapor  will  absorb  exactly  those  wave  lengths  which 
it  itself  is  capable  of  emitting  when  incandescent. 
This  is  for  precisely  the  same  reason  that  a 
tuning  fork  will  respond  to,  that  is,  absorb, 
only  vibrations  which  have  the  same  period 
as  those  which  it  is  itself  able  to  emit.  Since, 
then,  the  dark  line  in  the  yellow  portion  of 
the  sun's  spectrum  is  in  exactly  the  same  place 
as  the  bright  yellow  line  produced  by  incandes- 
cent sodium  vapor,  or  the  dark  line  which  is 
produced  whenever  white  light  shines  through 
sodium  vapor,  we  infer  that  sodium  vapor  must 
be  contained  in  the  sun's  atmosphere.  By  com- 
paring in  this  way  the  positions  of  the  lines 
in  the  spectra  of  different  elements  with  the 
positions  of  various  dark  lines  in  the  sun's 
spectrum,  many  of  the  elements  which  exist 
on  the  earth  have  been  proved  to  exist  also  in 
the  sun.  For  example,  the  German  physicist 
KirchhofT  showed  that  the  four  hundred  sixty 
bright  lines  of  iron  which  were  known  to  him 
were  all  exactly  matched  by  dark  lines  in  the 
solar  spectrum.  -Fig.  448  shows  a  copy  of  a 
photograph  of  a  portion  of  the  solar  spectrum 
in  the  middle,  and  the  corresponding  bright- 
line  spectrum  of  iron  each  side  of  it.  It  will 
be  seen  that  the  coincidence  of  bright  and 
dark  lines  is  perfect. 

491.  Doppler's  principle  applied  to  light  waves.    We 

have  seen  (see  Doppler's  principle,  §  398,  p.  321)  that  FIG  44g    Com_ 

the  effect  of  the  motion  of  a  sounding  body  toward  an  parison  of  solar 

observer  is  to  shorten  slightly  the  wave  length  of  the  and  iron  spectra 


SPECTRA  407 

note  emitted,  and  the  effect  of  motion  away  from  an  observer  is  to 
increase  the  wave  length.  Similarly,  when  a  star  is  moving  toward 
the  earth  each  particular  wave  length  emitted  will  be  slightly  less  than 
the  wave  length  of  the  corresponding  light  from  a  source  on  the  earth's 
surface.  Hence  in  this  star's  spectrum  all  the  lines  will  be  displaced 
slightly  toward  the  violet  end  of  the  spectrum.  If  a  star  is  moving 
away  from  the  earth,  all  its  lines  will  be  displaced  toward  the  red  end. 
From  the  direction  and  amount  of  displacement,  therefore,  we  can  cal- 
culate the  velocity  with  which  a  star  is  moving  toward  or  receding  from 
the  solar  system.  Observations  of  this  sort  have  shown  that  some  stars 
are  moving  through  space  toward  the  solar  system  with  a  velocity  of 
150  miles  per  second,  while  others  are  moving  away  with  almost  equal 
velocities.  The  whole  solar  system  appears  to  be  sweeping  through 
space  with  a  velocity  of  about  12  miles  per  second  ;  but  even  at  this  rate 
it  would  be  at  least  1,000,000  years  before  the  earth  would  come  into 
the  neighborhood  of  the  nearest  star,  even  if  it  were  moving  directly 
toward  it. 

QUESTIONS  AND  PROBLEMS 

1.  In  what  part  of  the  sky  will  a  rainbow  appear  if  it  is  formed  in 
the  early  morning? 

2.  Is  a  bow  seen  at  4  o'clock  in  the  afternoon  higher  or  lower  than 
a  bow  seen  at  5  o'clock  on  the  same  day  ? 

3.  Why  is  a  rainbow  never  seen  during  the  middle  part  of  the  day? 

4.  If  you  look  at  a  broad  sheet  of  white  paper  through  a  prism,  it  will 
appear  red  at  one  edge  and  blue  at  the  other,  but  white  in  the  middle. 
Explain  why  the  middle  appears  uncolored. 

5.  What  evidence  have  we  for  believing  that  there  is  sodium  in  the 
sun? 

6.  What  sort  of  a  spectrum  should  moonlight  give  ?    (The  moon  has 
no  atmosphere.) 

7.  If  you  were  given  a  mixture  of  a  number  of  salts,  how  would  you 
proceed  with  a  Bunsen  burner,  a  prism,  and  a  slit,  to  determine  whether 
or  not  there  was  any  calcium  in  the  mixture  ? 

8.  Can  you  see  any  reason  why  the  vibrating  molecules  of  an  incan- 
descent gas  might  be  expected  to  give  out  a  few  definite  wave  lengths, 
while  the  particles  of  an  incandescent  solid  give  out  all  possible  wave 
lengths  ? 

9.  Can  you  see  any  reason  why  it  is  necessary  to  have  the  slit  narrow 
and  the  slit  and  screen  at  conjugate  foci  of  the  lens  in  order  to  show 
the  Fraunhofer  lines  in  the  experiment  of  §  489  ? 


CHAPTER  XXI 


INVISIBLE  RADIATIONS 
RADIATION  FROM  A  HOT  BODY 

492.  Invisible  portions  of  the  spectrum.  When  a  spectrum 
is  photographed  the  effect  on  the  photographic  plate  is  found 
to  extend  far  beyond  the.  limits  of  the  shortest  visible  violet 
rays.  These  so-called  ultra-violet  rays  have  been  photographed 
and  measured  by  Lyman  of  Harvard  down  to  a  wave  length 
of  .00001  centimeter,  which  is  only  one  fourth 
the  wave  length  of  the  shortest  violet  waves. 

The  longest  rays  visible  in  the  extreme  red 
have  a  wave  length  of  about  .00008  centime- 
ter, but  delicate  thermoscopes  reveal  a  so-called 
infra-red  portion  of  the  spectrum,  the  investiga- 
tion of  which  was  carried  in  1912,  by  Rubens  and 
von  Bseyer  of  Berlin,  to  wave  lengths  as  long  as 
,03  centimeter,  400  times  as  long  as  the  longest 
visible  rays. 

The  presence  of  these  long  heat  rays  may  be  detected 
by  means  of  the  radiometer  (Fig.  449),  an  instrument 
perfected  by  E.  F.  Nichols  of  Dartmouth.  In  its  common 
form  it  consists  of  a  partially  exhausted  bulb,  within  which  is  a  little 
aluminium  wheel  carrying  four  vanes  blackened  on  one  face  and  polished 
on  the  other.  When  the  instrument  is  held  in  sunlight  or  before  a  lamp, 
the  vanes  rotate  in  such  a  way  that  the  blackened  faces  always  move 
away  from  the  source  of  radiation  because  they  absorb  ether  waves 
better  than  do  the  polished  faces,  and  thus  become  hotter.  The  heated 
air  in  contact  with  these  faces  then  exerts  a  greater  pressure  against 
them  than  does  the  air  in  contact  with  the  polished  faces.  The  more 
intense  the  radiation,  the  faster  is  the  rotation. 

408 


FIG.  449.  The 
Crookes  radi- 
ometer 


RADIATION  FROM  A  HOT  BODY 


409 


F  A 


A  still  simpler  way  of  studying  these  long  heat  waves  was  devised 
in  1912  by  Trowbridge  of  Princeton.  A  rubber  band  AC  (Fig.  450)  a 
millimeter  wide  is  stretched  to  double  its  length  over  a  glass  plate 
FGHlj  and  the  thinnest  possible  glass  staff  ED,  car- 
rying a  light  mirror  E  about  2  millimeters  square,  is 
placed  under  the  rubber  band  at  its  middle  point  B. 
When  the  spectrum  is  thrown  upon  the  portion  AB 
of  the  band,  the  change  in  its  length  produced  by 
the  heating  causes  ED  to  roll,  and  a  spot  of  light 
reflected  from  E  to  the  wall  to  shift  its  position 
by  an  amount  proportional  to  the  heating. 

Let  either  the  radiometer  or  the  thermoscope 
described  above  be  placed  just  beyond  the  red  end  of 
the  spectrum.  It  will  indicate  the  presence  of  heat 
rays  here  of  even  greater  energy  than  those  in  the 
visible  spectrum.  Again,  let  a  red-hot  iron  ball  and  one  of  the  detectors 
be  placed  at  conjugate  foci  of  a  large  mirror  (Fig.  451).  The  invisible 
heat  rays  will  be  found  to  be  reflected  and  focused  just  as  are  light 
rays.  Next  let  a  flat  bottle  filled  with  water  be  inserted  between  the 
detector  and  any  source  of  heat.  It  will  be  found  that  water,  although 
transparent  to  light  rays,  absorbs  nearly  all  of  the  infra-red  rays.  But 
if.  the  water  is  replaced  by  carbon  bisulphide,  the  infra-red  rays  will 
be  freely  transmitted, 
even  though  the  liquid 
is  rendered  opaque  to 
light  waves  by  dissolv- 
ing iodine  in  it. 


FIG.  450.  A  simple 
thermoscope 


FIG.  451.    Reflection  of  infra-red  rays 


493.  Radiation  and 
temperature.  All  bod- 
ies, even  such  as  are 
at  ordinary  tempera- 
tures, are  continually 

radiating  energy  in  the  form  of  ether  waves.  This  is  proved  by 
the  fact  that  even  if  a  body  is  placed  in  the  best  vacuum  ob- 
tainable, it  continually  falls  in  temperature  when  surrounded 
by  a  colder  body,  such,  for  example,  as  liquid  air.  The  ether 
waves  emitted  at  ordinary  temperatures  are  doubtless  very 
long  as  compared  with  light  waves.  As  the  temperature  is 


410  INVISIBLE  KADIATIONS 

raised,  more  and  more  of  these  long  waves  are  emitted,  but 
shorter  and  shorter  waves  are  continually  added.  At  about 
525°  C.  the  first  visible  waves,  that  is,  those  of  a  dull  red 
color,  begin  to  appear.  From  this  temperature  on,  owing  to 
the  addition  of  shorter  and  shorter  waves,  the  color  changes 
continuously  —  first  to  orange,  then  to  yellow,  and  finally, 
between  800°  C.  and  1200°  C.,  to  white.  In  other  words,  all 
bodies  get  "red-hot"  at  about  525°  C.  and  "white-hot"  at 
from  800°  C.  to  1200°  C. 

Some  idea  of  how  rapidly  the  total  radiation  of  ether  waves 
increases  with  increase  of  temperature  may  be  obtained  from 
the  fact  that  a  hot  platinum  wire  gives  out  thirty-six  times 
as  much  light  at  1400°  C.  as  it  does  at  1000°  C.,  although 
at  the  latter  temperature  it  is  already  white-hot.  The  radi- 
ations from  a  hot  body  are  sometimes  classified  as  heat 
rays,  light  rays,  and  chemical  or  actinic  rays.  The  classifi- 
cation is,  however,  misleading,  since  all  ether  waves  are  heat 
waves,  in  the  sense  that  when  absorbed  by  matter  they  pro- 
duce heating  effects ;  that  is,  molecular  motions.  Radiant 
heat  is,  then,  the  radiated  energy  of  ether  waves  of  any  and  all 
wave  lengths. 

494.  Radiation  and  absorption.  Although  all  substances 
begin  to  emit  waves  of  a  given  wave  length  at  approximately 
the  same  temperature,  the  total  rate  of  emission  of  energy  at  a 
given  temperature  varies  greatly  with  the  nature  of  the  radiat- 
ing surface.  In  general,  experiment  shows  that  surfaces  which 
are  good  absorbers  of  ether  radiations  are  also  good  radiators. 
From  this  it  follows  that  surfaces  which  are  good  reflectors,  like 
the  polished  metals,  must  be  poor  radiators. 

Thus,  let  two  sheets  of  tin,  5  or  10  centimeters  square,  one  brightly 
polished  and  the  other  covered  on  one  side  with  lampblack,  be  placed 
in  vertical  planes  about  10  centimeters  apart,  the  lampblacked  side  of 
one  facing  the  polished  side  of  the  other.  Let  a  small  ball  be  stuck 
with  a  bit  of  wax  to  the  outer  face  of  each.  Then  let  a  hot  metal 


BADIATION  FROM  A  HOT  BODY 


411 


plate  or  ball  (Fig.  452)  be  held  midway  between  the  two.  The  wax  on 
the  tin  with  the  blackened  face  will  melt  and  its  ball  will  fall  first, 
showing  that  the  lampblack  ab- 
sorbs the  heat  rays  faster  than 
does  the  polished  tin.  Now 
let  two  blackened  glass  bulbs 
be  connected,  as  in  Fig.  453, 
through  a  U-tube  containing 
colored  water,  and  let  a  well- 
polished  tin  can,  one  side  of 
which  has  been  blackened,  be 
filled  with  boiling  water  and 
placed  between  them.  The  mo- 
tion of  the  water  in  the  U-tube 
will  show  that  the  blackened  side  of  the  can  is  radiating  heat  much  more 
rapidly  than  the  other,  although  the  two  are  at  the  same  temperature. 


FIG.  452.  Good  re- 
flectors  are    poor 
absorbers 


FIG.  453.  Good  ab- 
sorbers   are    good 
radiators 


QUESTIONS  AND  PROBLEMS 

1.  When  one  is  sitting  in  front  of  an  open  grate  fire  does  he  receive 
most  heat  by  conduction,  convection,  or  radiation? 

2.  The  atmosphere  is  transparent  to  most  of  the  sun's  rays.    Why 
are  the  upper  regions  of  the  atmosphere  so  much  colder  than  the  lower 
regions  ? 

3.  Sunlight  in  coming  to  the  eye  travels  a  much  longer  air  path 
at  sunrise  and  sunset  than  it  does  at  noon.    Since  the   sun  appears 
red  or  yellow  at  these  times,  what  rays  are   absorbed  most  by  the 
atmosphere  ? 

4.  Glass  transmits  all  the  visible  waves,  but  does  not  transmit  the 
long  infra-red  rays.    Hence  explain  the  principle  of  the  hotbed. 

5.  Which  will  be  cooler  on  a  hot  day,  a  white  hat  or  a  black  one? 

6.  Will  tea  cool  more  quickly  in  a  polished  or  in  a  tarnished  metal 
vessel ? 

7.  Which  emits  the  more  red  rays,  a  white-hot  iron  or  the  same  iron 
when  it  is  red-hot? 

8.  Liquid  air  flasks  and  thermos  bottles  are  doubled-walled  glass 
vessels  with  a  vacuum  between  the  walls.    Liquid  air  will  keep  many 
times  longer  if  the  glass  walls  are  silvered  than  if  they  are  not.    Why? 
Why  is  the  space  between  the  walls  evacuated  ? 


412 


INVISIBLE  RADIATIONS 


ELECTRICAL  RADIATIONS 

495.  Proof  that  the  discharge  of  a  Leyden  jar  is  oscillatory. 

We  found  in  §  419,  p.  340,  that  the  sound  waves  sent  out 
by  a  sounding  tuning  fork  will  set  into  vibration  an  adjacent 
fork,  provided  the  latter  has  the  same  natural  period  as  the 
former.  The  following  is  the  complete  electrical  analogy  of 
this  experiment. 

Let  the  inner  and  outer  coats  of  a  Leyden  jar  A  (see  Fig.  454)  be 
connected  by  a  loop  of  wire-  cdcf,  the  sliding  crosspiece  de  being  arranged 
so  that  the  length  of  the  loop  may  be  altered  at  will.  Also  let  a  strip 
of  tin  foil  be  brought  over  the  edge  of  this  jar  from  the  inner  coat  to 
within  about  1  millimeter 
of  the  outer  coat  at  C.  Let 
the  two  coats  of  an  exactly 
similar  jar  B  be  connected 
with  the  knobs  n  and  n'  by 
a  second  similar  wire  loop 
of  fixed  length.  Let  the 
two  jars  be  placed  side  by 

side  with  their  loops  par-      -~  ,     M 

FIG.  454.    Sympathetic  electrical  vibrations 
allel,  and  let  the  jar  B  be 

successively  charged  and  discharged  by  connecting  its  coats  with  a 
static  machine  or  an  induction  coil.  At  each  discharge  of  jar  />  through 
the  knobs  n  and  n'  a  spark  will  appear  in  the  other  jar  at  C,  provided 
the  crosspiece  de  is  so  placed  that  the  areas  of  the  two  loops  are  equal. 
When  de  is  slid  along  so  as  to  make  one  loop  considerably  larger  or 
smaller  than  the  other,  the  spark  at  C  will  disappear. 

The  experiment  therefore  demonstrates  that  two  electrical 
circuits,  like  two  tuning  forks,  can  be  tuned  so  as  to  respond  to 
each  other  sympathetically,  and  that  just  as  the  tuning  forks 
will  cease  to  respond  as  soon  as  the  period  of  one  is  slightly 
altered,  so  this  electric  resonance  disappears  when  the  exact 
symmetry  of  the  two  circuits  is  destroyed.  Since,  obviously, 
this  phenomenon  of  resonance  can  occur  only  between  systems 
which  have  natural  periods  of  vibration,  the  experiment  proves 
that  the  discharge  of  a  Leyden  jar  is  a  vibratory,  that  is,  an 


ELECTRICAL  RADIATIONS  413 

oscillatory,  phenomenon.  As  a  matter  of  fact,  when  such  a 
spark  is  viewed  in  a  rapidly  revolving  mirror  it  is  actually  found 
to  consist  of  from  ten  to  thirty  flashes  following  each  other  at 
equal  intervals.  Fig.  455  is  a  photograph  of  such  a  spark. 

In  spite  of  these  oscillations  the  whole  discharge  may  be 
made  to  take  place  in  the  incredibly  short  time  of  1  0  0  bV  o  o  o 
of  a  second.  This  fact  coupled 
with  the  extreme  brightness  of 
the  spark  has  made  possible  the 
surprising  results  of  so-called 

instantaneous  electric-spark  pho- 

7          rm  .,  FIG.  455.    Oscillations  of  the 

tography.    The    plate    opposite  electric  spark 

page  414  shows  the  passage  of 

a  bullet  through  a  soap  bubble.  The  film  was  rotated  contin- 
uously instead  of  intermittently,  as  in  ordinary  moving-picture 
photography.  The  illuminating  flashes,  5000  per  second,  were 
so  nearly  instantaneous  that  the  outlines  are  not  blurred. 

496.  Electric  waves.  The  experiment  of  §  495  demonstrates 
not  only  that  the  discharge  of  a  Ley  den  jar  is  oscillatory,  but 
also  that  these  electrical  oscillations  set  up  in  the  surrounding 
medium  disturbances,  or  waves  of  some  sort,  which  travel  to  a 
neighboring  circuit  and  act  upon  it  precisely  as  the  air  waves 
acted  on  the  second  tuning  fork  in  the  sound  experiment. 
Whether  these  are  waves  in  the  air,  like  sound  waves,  or  dis- 
turbances in  the  ether,  like  light  waves,  can  be  determined  by 
measuring  their  velocity  of  propagation.  The  first  determina- 
tion of  this  velocity  was  made  by  Heinrich  Hertz  in  Gerniany 
in  1888.  He  found  it  to  be  precisely  the  same  as  that  of  light ; 
that  is,  300,000  kilometers  per  second.  This  result  shows,  there- 
fore, that  electrical  oscillations  set  up  waves-  in  the  ether.  These 
waves  are  now  known  as  Hertz  waves. 

The  length  of  the  waves  emitted  by  the  oscillatory  spark 
of  instantaneous  photography  is  evidently  very  great,  namely, 
about  \7o7o!  ooV  =  30  meters'  since  the  velocity  of  light  is 


414  INVISIBLE  RADIATIONS 

300,000,000  meters  per  second,  and  since  there  are  10,000,000 
oscillations  per  second;  for  we  have  seen  in  §  393,  p.  318, 
that  wave  length  is  equal  to  velocity  divided  by  the  number 
of  oscillations  per  second.  By  diminishing  the  size  of  the  jar 
and  the  length  of  the  circuit  the  length  of  the  waves  may  be 
greatly  reduced.  By  causing  the  electrical  discharges  to  take 
place  between  two  balls  only  a  fraction  of  a  millimeter  in 
diameter,  instead  of  between  the  coats  of  a  condenser,  elec- 
trical waves  have  been  obtained  as  short  as  .3  centimeter, 
only  ten  times  as  long  as  the  longest  measured  heat  waves. 

497.  The  coherer.    In  the  experiment  of  §  495  we  detected 
the  presence  of  the  electrical  waves  by  means  of  a  small  spark 
gap   C  in  a  circuit  almost  identical  with  that  in  which  the 
oscillations  were  set  up.    This  same  means  may  be  employed 
for  the  detection  of  waves  many  feet  away  from  the  source, 
but  the  instrument  with  which  electromagnetic  waves  were 
first  detected  hundreds  of  miles  away  from  the  source  was  the 
coherer.  Its  principle  is  illustrated  in  the  following  experiment: 

Let  a  glass  tube  several  centimeters  long  and  6  or  8  millimeters  in 
diameter  be  filled  with  fine  brass  or  nickel  filings,  and  let  copper  wires 
be  thrust  into  these  filings  within  a  distance  of  about  a  centimeter  of 
each  other.  Let  these  wires  be  connected  in  series  with  a  Daniell  cell 
and  a  simple  D'Arsonval  galvanometer.  The  resistance  of  the  loose 
contacts  of  the  filings  will  be  so  great  that  very  little  current  will  flow 
through  the  circuit.  Now  let  a  static  machine  be  started  many  feet 
away.  The  galvanometer  will  show  a  strong  deflection  as  soon  as  a  spark 
passes  between  the  knobs  of  the  electrical  machine.  This  is  because  the 
electric  waves,  as  soon  as  they  fall  upon  the  filings,  cause  them  to  cohere 
or  cling  together,  so  that  the  electrical  resistance  of  the  tube  of  filings  is 
reduced  to  a  small  fraction  of  what  it  was  before.  If  the  tube  is  tapped 
with  a  pencil,  the  old  resistance  will  be  restored,  because  the  filings  have 
been  broken  apart  by  the  jar.  The  experiment  may  then  be  repeated. 

498.  Wireless  telegraphy.    The  last  experiment  illustrates  completely 
the  method  of  transmitting  wireless  messages  during  the  first  decade 
after  Marconi,  in  1896,  had  realized  commercial  wireless  telegraphy. 
At  present  the  essential  elements  of  the  Marconi  system  of  wireless 


CINEMATOGRAPH  FILM  OF  A  BULLET  FIRED  THROUGH  A  SOAP  BUBBLE 

The  flight  of  the  missile  may  be  followed  easily.   It  will  be  seen  that  the  bubble 

breaks,  not  when  the  bullet  enters,  but  when  it  emerges.  (From"  Moving  Pictures," 

by  F.  A.  Talbot.  Courtesy  of  J.  B.  Lippincott  Company) 


ELECTRICAL  RADIATIONS 


415 


telegraphy  are  as  follows  :  The  transmitter  consists  of  an  ordinary  induc- 
tion coil  or  transformer  771,  through  the  primary  of  which  [Fig.  456,  (1)] 
a  current  is  sent  from  the  alternator  A.  The  secondary  $  of  this  trans- 
former charges  the  condenser  C1  until  its  potential  rises  high  enough  to 
cause  a  spark  discharge  to  take  place  across  the  gap  s.  This  discharge 
of  C\  is  oscillatory  (§  495),  the  frequency  being  of  the  order  of  1,000,000 
per  second,  but  subject  to  the  control  of  the  operator  through  the  slid- 
ing contacts  c,  precisely  as  in  the  case  illustrated  in  Fig.  454.  The 


FIG.  456.    Transmitting  and  receiving  stations  for  wireless  telegraphy 
(1)  Transmitting  station;  (2)  receiving  station 

oscillations  in  this  condenser  circuit  induce  oscillations  in  the  aerial-wire 
system,  which  is  tuned  to  resonance  with  it  through  the  sliding  contact  c'* 
The  waves  sent  out  by  this  aerial  system  induce  like  oscillations  in 
the  aerial  system  of  the  receiving  station  [Fig.  456,  (2)],  it  may  be  thou- 
sands of  miles  away,  which  is  tuned  to  resonance  with  it  through  the 
variable  capacity  C2  and  "  inductance  "  B.  These  oscillations  induce 
exactly  similar  ones  in  the  condenser  circuits  /1?  /2,  and  /3,  all  of  which 
are  tuned  to  resonance  with  the  receiving  aerial  system.  The  detector 
of  the  oscillations  in  73  is  simply  a  crystal  of  carborundum  D  in  series 
with  a  telephone  receiver  R.  This  crystal,  like  the  mercury  arc  of  §  384, 

*  In  the  diagram  an  arrow  drawn  diagonally  across  a  condenser  indicates  that 
for  the  sake  of  tuning  the  condenser  is  made  adjustable.  Similarly,  an  arrow 
across  two  circuits  coupled  inductively,  like  the  primary  and  secondary  of  the 
"  oscillation  transformer/'  T2,  indicates  that  the  amount  of  interaction  of  the  two 
circuits  can  be  varied,  as,  for  example,  by  sliding  one  coil  a  larger  or  smaller 
distance  inside  the  other. 


416  INVISIBLE  RADIATIONS 

has  the  property  of  transmitting  a  current  in  one  direction  only.  Were 
it  not  for  this  property  the  telephone  could  not  be  used  as  a  detector 
because  the  frequency  is  so  high  —  of  the  order  of  a  million.  In  view 
of  this  property,  however,  while  the  oscillations  of  a  given  spark  last,  an 
intermittent  current  passes  in  one  direction  and  then  ceases  until  the 
oscillations  of  the  next  spark  arrive.  Since  from  300  to  1000  sparks  pass 
at  .s  per  second  when  the  key  K  is  closed,  the  operator  hears  a  musical 
note  of  this  frequency  as  long  as  K  is  depressed.  Long  and  short  notes 
then  correspond  to  the  dots  and  dashes  of  ordinary  telegraphy. 

The  stretching  of  the  aerial  wires  horizontally  instead  of  vertically, 
as  was  formerly  done,  permits  to  some  extent  of  directive  sending  and 
receiving,  for  as  in  the  experiment  of  §  495  the  sending  and  receiving 
wires  work  best  when  they  are  parallel. 

The  three  tuned  circuits,  Ilt  72,  /3,  are  used  because  such  a  series  of 
tuned  circuits  does  not  pick  up  waves  of  other  periods.  For  "  nonselec- 
tive  "  receiving  these  circuits  are  omitted  and  the  detector  and  tele- 
phone are  placed  directly  across  the  condenser  (72.  The  resistance  of 
the  telephone  is  so  high  that  it  does  not  interfere  with  the  oscillations 
of  the  condenser  system  across  which  it  is  placed. 

499.  The  electromagnetic  theory  of  light.  The  study  of 
electromagnetic  radiations,  like  those  discussed  in  the  pre- 
ceding paragraphs,  has  shown  not  only  that  they  have  the 
speed  of  light,  but  that  they  are  reflected,  refracted,  and 
polarized ;  in  fact,  that  they  possess  all  the  properties  of  light 
waves,  the  only  apparent  difference  being  in  their  greater 
wave  length.  Hence  modern  physics  regards  light  as  an  electro- 
magnetic phenomenon ;  that  is,  light  waves  are  thought  to  be 
generated  by  the  oscillations  of  the  electrically  charged  parts 
of  the  atoms.  It  was  as  long  ago  as  1864  that  Clerk-Maxwell, 
of  Cambridge,  England,  one  of  the  world's  most  brilliant 
physicists  and  mathematicians,  showed  that  it  ought  to  be 
possible  to  create  ether  waves  by  means  of  electrical  disturb- 
ances. But  the  experimental  confirmation  of  his  theory  did 
not  come  until  the  time  of  Hertz's  experiments  (1888). 
Maxwell  and  Hertz  together,  therefore,  share  the  honor  of 
establishing  the  modern  electromagnetic  theory  of  light  (p.  54). 


CATHODE  AND  EONTGEN  RAYS  417 

CATHODE  AND  RONTGEN  RAYS 

500.  The  electric  spark  in  partial  vacua.   Let  a  and  b  (Fig.  457) 

be  the  terminals  of  an  induction  coil  or  static  machine,  e  and  /  electrodes 
sealed  into  a  glass  tube  60  or  80  centimeters  long,  g  a  rubber  tube  lead- 
ing to  an  air  pump  by  which  the  tube  may  be  exhausted.  Let  the  coil 
be  started  before  the  exhaustion  is  begun.  A  spark  will  pass  between  a 
and  b,  since  ab  is  a  very  much  shorter  path  than  ef.  Then  let  the  tube  be 
rapidly  exhausted.  When  the  pressure  has  been  reduced  to  a  few  centi- 
meters of  mercury  the  discharge  will  be  seen  to  choose  the  long  path  efiu 
preference  to  the  short  path  ab, 
thus  showing  that  an  electrical 
discharge  takes  place  more  read- 
ily through  a  partial  vacuum  than 
through  air  at  ordinary  pressures. 

9 

When  the  spark  first  be- 

,        FIG.  457.    Discharge  in  partial  vacua 
gins  to  pass  between  e  and 

/;  it  will  have  the  appearance  of  a  long  ribbon  of  crimson  light. 
As  the  pumping  is  continued  this  ribbon  will  spread  out  until 
the  crimson  glow  fills  the  whole  tube.  Ordinary  so-called 
Geissler  tubes  are  tubes  precisely  like  the  above,  except  that 
they  are  usually  twisted  into  fantastic  shapes,  and  are  some- 
times surrounded  with  jackets  containing  colored  liquids, 
which  produce  pretty  color  effects. 

501.  Cathode  rays.   When  a  tube  like  the  above  is  exhausted 
to  a  very  high  degree,  say,  to  a  pressure  of  about  .001  milli- 
meter  of  mercury,   the  character   of  the   discharge   changes 
completely.    The  glow  almost  entirely  disappears  from  the 
residual  gas  in  the  tube,  and  certain  invisible  radiations  called 
cathode  rays  are  found  to  be  emitted  by  the  cathode  (the 
terminal  of  the  tube  which  is  connected  to  the  negative  ter- 
minal of  the  coil  or  static  machine).    These  rays  manifest 
themselves  first  by  the  brilliant  fluorescent  effects  which  they 
produce  in  the  glass  walls  of  the  tube,  or  in  other  substances 
within  the  tube  upon  which  they  fall ;  second,  by  powerful  heat- 
ing effects ;  and  third,  by  the  sharp  shadows  which  they  cast. 

t 


418 


INVISIBLE  RADIATIONS 


Thus,  if  the  negative  electrode  is  concave,  as  in  Fig.  458,  and  a  piece 
of  platinum  foil  is  placed  at  the  center  of  the  sphere  of  which  the  cathode 
is  a  portion,  the  rays  will  come  to  a  focus  upon  a 
small  part  of  the  foil  and  will  heat  it  white-hot,  thus 
showing  that  the  rays,  whatever  they  are,  travel  out 
in  straight  lines  at  right  angles  to  the  surface  of 
the  cathode.  This  may  also  be  shown  nicely  by  an 
ordinary  bulb  of  the  shape  shown  in  Fig.  460.  If  the 
electrode  A  is  made  the  cathode  and  B  the  anode,  a 
sharp  shadow  of  the  piece  of  platinum  in  the  middle 
of  the  tube  will  be  cast  on  the  wall  opposite  to  A,  thus 
showing  that  the  cathode  rays,  unlike  the  ordinary 
electric  spark,  do  not  pass  between  the  terminals  of 
the  tube,  but  pass  out  in  a  straight  line  from  the 
cathode  surface. 


FIG.  458.   Heating 

effect    of   cathode 

rays 


502.  Nature  of  the  cathode  rays.  The  na- 
ture of  the  cathode  rays  was  a  subject  of  much  dispute  between 
the  years  1875,  when  they  first  began  to  be  carefully  studied, 
and  1898.  Some  thought  them  to  be  streams  , 

of  negatively  charged  particles  shot  off  with 
great  speed  from  the  surface  of  the  cathode, 
while  others  thought  they  were  waves  in  the 
ether  —  some  sort  of  invisible  light.  The  fol- 
lowing experiment  furnishes  very  convincing 
evidence  that  the  first  view  is  correct. 

NP  (Fig.  459)  is  an  exhausted  tube  within  which 
has  been  placed  a  screen  sf  coated  with  some  sub- 
stance like  zinc  sulphide,  which  fluoresces  brilliantly 
when  the  cathode  rays  fall  upon  it ;  mn  is  a  mica 
strip  containing  a  slit  s.  This  mica  strip  absorbs  all 
the  cathode  rays  which  strike  it;  but  those  which 
pass  through  the  slit  s  travel  the  full  length  of  the 
tube,  and  although  they  are  themselves  invisible,  their 
path  is  completely  traced  out  by  the  fluorescence 
which  they  excite  upon  ,s/as  they  graze  along  it.  If  a 
magnet  M  is  held  in  the  position  shown,  the  cathode 
rays  will  be  seen  to  be  deflected,  and  in  exactly  the  direction  to  be 
expected  if  they  consisted  of  negatively  charged  particles.  For  we 


FIG.  459.  Deflec- 
tion of  cathode 
rays  by  a  magnet 


CATHODE  AND  RONTGEN  RAYS 


419 


learned  in  §  308,  p.  240,  that  a  moving  charge  constitutes  an  electric 
current,  and  in  §  360,  p.  287,  that  an  electric  current  tends  to  move  in 
an  electric  field  in  the  direction  given  by  the  motor  rule.  On  the  other 
hand,  a  magnetic  field  is  not  known  to  exert  any  influence  whatever  on 
the  direction  of  a  beam  of  light  or  on  any  other  form  of  ether  waves. 

When,  in  1895,  J.  J.  Thomson,  of  Cambridge,  England, 
proved  that  the  cathode  rays  were  also  deflected  by  electric 
charges,  as  was  to  be  expected  if  they  consist  of  negatively 
charged  particles,  and  when  Perriii  in  Paris  had  proved  that 
they  actually  impart  negative  charges  to  bodies  on  which  they 
fall,  all  opposition  to  the  projected-particle  theory  was  aban- 
doned. The  mass  and  speed  of  these  particles  are  computed 
from  their  deflectibility  in  magnetic  and  electric  fields. 

Cathode  rays  are  then  to-day  universally  recognized  as  streams 
of  electrons  shot  off  from  the  surface  of  the  cathode  with  speeds 
tvhich  may  reach  the  stupendous  value  o/"  100,000  miles  per  second. 

503.  X  rays.  It  was  in  1895  that  the  German  physicist, 
Rontgen,  first  discovered  that  wherever  the  cathode  rays  im- 
pinge upon  the  walls  of  a  tube,  or  upon  any  obstacles  placed 
inside  the  tube,  they  give  rise  to  another  type  of  invisible 
radiation  which  is  now  ^"V !","""///, 

known  under  the  name  of 
X  rays,  or  Rontgen  rays. 
In  the  ordinary  X-ray  tube 
(Fig.  460)  a  thick  piece 
of  platinum  P  is  placed 
in  the  center  to  serve  as 
a  target  for  the  cathode 
rays,  which,  being  emitted  at  right  angles  to  the  concave  sur- 
face of  the  cathode  (7,  come  to  a  focus  at  a  point  on  the 
surface  of  this  plate.  This  is  the  point  at  which  the  X  rays 
are  generated  and  from  which  they  radiate  in  all  directions. 

In  order  to  convince  oneself  of  the  truth  of  this  statement,  it  is  only 
necessary  to  observe  an  X-ray  tube  in  action.  It  will  be  seen  to  be 


FIG.  460.    An  X-ray  bulb 


420  INVISIBLE  RADIATIONS 

divided  into  two  hemispheres  by  the  plane  which  contains  the  platinum 
plate  (see  Fig.  460).  The  hemisphere  which  is  facing  the  source  of  the 
X  rays  will  be  aglow  with  a  greenish  fluorescent  light,  while  the  other 
hemisphere,  being  screened  froin  the  rays,  is  darker.  By  moving  a 
fluoroscope  (a  zinc  sulphide  screen)  about  the  tube  it  will  be  made  evident 
that  the  rays  which  render  the  bones  visible  (Fig.  461)  come  from  P. 

504.  Nature  of  X  rays.    While  X  rays  are  like  cathode  rays 
in  producing  fluorescence,  and  also  in  that  neither  of  them  can 
be  reflected,  refracted,  or  polarized,  as  is  light,  they- nevertheless 
differ  from  cathode  rays  in  several  important  respects.    First, 
X  rays  penetrate  many  substances  which  are  quite  impervious 
to  cathode  rays ;  for  example,  they  pass  through  the  walls  of 
the  glass  tube,  while  cathode  rays  ordinarily  do  not.    Again, 
X  rays  are  not  deflected  either  by  a  magnet  or  by  an  electro- 
static charge,  nor  do  they  carry  electrical  charges  of  any  sort. 
Hence  it.  is  certain  that  they  do  not  consist,  like  cathode  rays, 
of  streams  of  electrically  charged  particles.    Their  real  nature 
is  still  unknown,  but  they  are  at  present  generally  regarded  as 
irregular  pulses  in  the  ether,  set  up  by  the  sudden  stopping  of 
the  cathode-ray  particles  when  they  strike  an  obstruction. 

505.  X  rays  render  gases  conducting.    One  of  the  notable 
properties  which  X  rays  possess  in  common  with  cathode  rays 
is  the  property  of  causing  any  electrified  body  on  which  they 
fall  to  slowly  lose  its  charge. 

To  demonstrate  the  existence  of  this  property,  let  any  X-ray  bulb  be 
set  in  operation  within  5  or  10  feet  of  a  charged  gold-leaf  electroscope. 
The  leaves  at  once  begin  to  fall  together. 

The  reason  for  this  is  that  the  X  rays  shake  loose  electrons 
from  the  atoms  of  the  gas  and  thus  fill  it  with  positively  and 
negatively  charged  particles,  each  negative  particle  being  at  the 
instant  of  separation  an  electron,  and  each  positive  particle  an 
atom  from  which  an  electron  has  been  detached.  Any  charged 
body  in  the  gas  therefore  draws  toward  itself  charges  of  sign 
opposite  to  its  own,  and  thus  becomes  discharged. 


JOSEPH  JOHN  THOMSON  (1856-        ) 

Most  conspicuous  figure  in  the  development  of  the  "physics  of  the  electron"  ; 
born  in  Manchester,  England;  educated  at  Cambridge  University;  Cavendish 
professor  of  experimental  physics  in  Cambridge  since  1884 ;  author  of  a  number  of 
books,  the  most  important  of  which  is  the  "Conduction  of  Electricity  through 
Gases,"  1903;  author  or  inspirer  of  much  of  the  recent  work,  both  experimental 
and  theoretical,  which  has  thrown  light  upon  the  connection  between  electricity 
and  matter ;  worthy  representative  of  twentieth-century  physics 


EADIOACTIVITY 


421 


FIG.  461.   An  X-ray 

picture  of   a   living 

hand 


506.  X-ray  pictures.    The  most  striking  property  of  X  rays 
is  their  ability  to  pass  through  many  substances  which  are 
wholly  opaque  to  light,  such,  for  example,  as  cardboard,  wood, 
leather,  flesh,  etc.    Thus,  if  the  hand  is  held 

close  to  a  photographic  plate  and  then  ex- 
posed to  X  rays,  a  shadow  picture  of  the 
denser  portions  of  the  hand,  that  is,  the 
bones,  is  formed  upon  the  plate.  Fig.  461 
shows  a  copy  of  such  a  picture. 

RADIOACTIVITY 

507.  Discovery  of  radioactivity.    In  1896 
Henri  Becquerel,  in  Paris,  performed  the 
following  experiment.   He  wrapped  a  photo- 
graphic plate  in  a  piece  of  perfectly  opaque 
black  paper,  laid  a  coin  on  top  of  the  paper, 
and  suspended  above  the  coin  a  small  quan- 
tity of  the  mineral  uranium.    He  then  set  the  whole  away 
in  a  dark  room  and  let  it  stand  for  several  days.    When  he 
developed  the  photographic  plate  he  found  upon  it  a  shadow 
picture  of  the  coin  similar  to  an  X-ray  picture.    He  concluded, 
therefore,  that  uranium  possesses  the  property  of  spontaneously 
emitting  rays  of  some  sort  which  have  the  power  of  penetrat- 
ing opaque  objects  and  of  affecting  photographic  plates,  just  as 
X  rays  do.    He  also  found  that  these  rays,  which  he  called 
uranium  rays,  are  like  X  rays  in  that  they  discharge  electri- 
cally charged  bodies  on  which  they  fall.    He  found  also  that 
the  rays  are  emitted  by  all  uranium  compounds. 

508.  Radium.    It  was  but  a  few  months  after  Becquerel's 
discovery  that  Madame  Curie,  in  Paris,  began  an  investigation 
of  all  the  known  elements,  to  find  whether  any  of  the  rest  of 
them  possessed  the  remarkable  property  which  had  been  found 
to  be  possessed  by  uranium.    She  found  that  one  of  the  remain- 
ing known  elements,  namely  thorium,  the  chief  constituent 


422.  INVISIBLE  RADIATIONS 

of  Welsbach  mantles,  is  capable,  together  with  its  compounds, 
of  producing  the  same  effect.  After  this  discovery  the  rays 
from  all  this  class  of  substances  began  to  be  called  Becquerel 
rays,  and  all  substances  which  emitted  such  rays  were  called 
radioactive  substances. 

But  in  connection  with  this  investigation  Madame  Curie 
noticed  that  pitchblende,  the  crude  ore  from  which  uranium 
is  extracted,  and  which  consists  largely  of  uranium  oxide, 
would  discharge  her  electroscope  about  four  times  as  fast  as 
pure  uranium.  She  inferred,  therefore,  that  the  radioactivity 
of  pitchblende  could  not  be  due  solely  to  the  uranium  con- 
tained in  it,  and  that  pitchblende  must  therefore  contain  some 
hitherto  unknown  element  which  has  the  property  of  emitting 
Becquerel  rays  more  powerfully  than  uranium  or  thorium. 
After  a  long  and  difficult  search  she  succeeded  in  separating 
from  several  tons  of  pitchblende  a  few  hundredths  of  a  gram 
of  a  new  element  which  was  capable  of  discharging  an  electro- 
scope more  than  a  million  times  as  rapidly  as  either  uranium 
or'  thorium.  She  named  this  new  element  radium. 

509.  Nature  of  Becquerel  rays.  That  these  rays  which  are 
spontaneously  emitted  by  radioactive  substances  are  not  X 
rays,  in  spite  of  their  similarity  in  affecting  a  photographic 
plate,  in  causing  fluorescence,  and  in  discharging  electrified 
bodies,  is  proved  by  the  fact  that  they  are  found  to  be  deflected 
by  both  magnetic  and  electric  fields,  and  by  the  further  fact 
that  they  impart  electric  charges  to  bodies  upon  which  they  fall. 
These  properties  constitute  strong  evidence  that  radioactive 
substances  project  from  themselves  electrically  charged  particles. 

But  an  experiment  performed  in  1899  by  Rutherford,  then 
of  McGill  University,  Montreal,  showed  that  Becquerel  rays 
are  complex,  consisting  of  three  different  types  of  radiation, 
which  he  named  the  alpha,  the  beta,  and  the  gamma  rays.  The 
beta  rays  are  found  to  be  identical  in  all  respects  with  cath- 
ode rays,  that  is,  they  are  streams  of  electrons  projected  with 


RADIOACTIVITY  423 

velocities  varying  from  60,000  to  180,000  miles  per  second. 
The  alpha  rays  are  distinguished  from  these  by  their  very  much 
smaller  penetrating  power,  by  their  very  much  greater  power 
of  rendering  gases  conductors,  by  their  very  much  smaller 
deflectability  in  magnetic  and  electric  fields,  and  by  the  fact 
that  the  direction  of  the  deflection  is  opposite  to  that  of  the  beta 
rays.  From  this  last  fact,  discovered  by  Rutherford  in  1903, 
the  conclusion  is  drawn  that  the  alpha  rays  consist  of  positively 
charged  particles ;  and  from  the  amount  of  their  deflectability 
their  mass  has  been  calculated  to  be  about  four  times  that  of 
the  hydrogen  atom,  that  is,  about  7000  times  the  mass  of  the 
electron,  and  their  velocity  to  be  about  20,000  miles  per  second. 
Rutherford  and  Boltwood  have  collected  the  alpha  particles 
in  sufficient  amount  to  identify  them  definitely  as  positively 
charged  atoms  of  helium. 

The  difference  in  the  sizes  of  the  alpha  and  beta  particles 
explains  why  the  latter  are  so  much  more  penetrating  than  the 
former,  and  why  the  former  are  so  much  more  efficient  than  the 
latter  in  knocking  electrons  out  of  the  molecules  of  a  gas  and 
rendering  it  conducting.  A  sheet  of  aluminium  foil  .005  centi- 
meter thick  cuts  off  completely  the  alpha  rays,  but  offers  practi- 
cally no  obstruction  to  the  passage  of  the  beta  and  gamma  rays. 

The  gamma  rays  are  very  much  more  penetrating  even  than 
the  beta  rays,  and  are  not  at  all  deflected  by  magnetic  or  electric 
fields.  They  are  commonly  supposed  to  be  X  rays  produced 
by  the  impact  of  the  beta  particles  on  surrounding  matter. 

510.  Crookes's  spinthariscope.  In  1903  Sir  William  Crookes  devised  a 
little  instrument,  called  the  spinthariscope,  which  furnishes  very  direct 
and  striking  evidence  that  particles  are  being  continuously  shot  off  from 
radium  with  enormous  velocities.  In  the  spinthariscope  a  tiny  speck  of 
radium  R  (Fig.  462)  is  placed  about  a  millimeter  above  a  zinc  sulphide 
screen  S,  and  the  latter  is  then  viewed  through  a  lens  L,  which  gives 
from  ten  to  twenty  diameters  magnification.  The  continuous  soft  glow 
of  the  screen,  which  is  all  one  sees  with  the  naked  eye,  is  resolved  by 
the  microscope  into  hundreds  of  tiny  flashes  of  light.  The  appearance  is 


424  INVISIBLE  RADIATIONS 

as  though  the  screen  were  being  fiercely  bombarded  by  an  incessant 
rain  of  projectiles,  each  impact  being  marked  by  a  flash  of  light,  just 
as  sparks  fly  from  a  flint  when  struck  with  steel.  The  experiment  is  a 
very  beautiful  one,  and  it  furnishes  very  direct  and 
convincing  evidence  that  radium  is  continually  pro- 
jecting particles  from  itself  at  stupendous  speeds. 
The  flashes  are  due  to  the  impacts  of  the  alpha,  not 
the  beta,  particles  against  the  zinc  sulphide  screen. 


511.  Photographing  the  tracks  of  alpha 
and  beta  rays.    In  1912  C.  T.  R.  Wilson,  of 

Cambridge,  England,  succeeded  in  actually  FIG.  462.  Crookes's 
photographing,  with  the  aid  of  the  electric  spinthariscope 
spark  (see  §  495),  the  tracks  of  alpha  and  beta  particles  as  they 
shoot  through  air.  Some  of  his  photographs  are  reproduced 
in  the  frontispiece.  The  white  streaks  there  shown  are  directly 
due  to  water  vapor  condensed  upon  ions  formed  along  the 
paths  of  the  rays.  In  the  case  of  the  alpha  particles  the  little 
water  drops  are  so  close  together  that  the  photograph  shows 
a  continuous  white  streak.  This  is  on  account  of  the  tre- 
mendous ionizing  power  of  the  alpha  particle.  In  the  case  of 
the  beta  rays  the  ionization  is  relatively  feeble,  and  the  paths 
are  not  so  straight,  both  of  which  results  are  to  be  expected 
from  the  smallness  of  the  electron  (beta  particle)  in  compari- 
son with  the  helium  atom  (alpha  particle).  The  photograph 
obtained  when  an  X-ray  beam  was  passed  through  the  gas 
shows  that  the  effect  of  the  X  ray  is  to  eject  electrons  from  the 
molecules  of  the  gas.  The  path  of  each  of  these  ejected  elec- 
trons may  be  easily  traced  in  the  figure. 

512.  The  disintegration  of  radioactive  substances.   Whatever 
be  the  cause  of  this  ceaseless  emission  of  particles  exhibited 
by  radioactive  substances,  it  is  certainly  not  due  to  any  ordi- 
nary chemical  reactions ;  for  Madame  Curie  showed,  when  she 
discovered  the  activity  of  thorium,  that  the  activity  of  all  the 
radioactive  substances  is  simply  proportional  to  the  amount 
of  the  active  element  present,  and  has  nothing  whatever  to  do 


RADIOACTIVITY  425 

with  the  nature  of  the  chemical  compound  in  which  the  ele- 
ment is  found.  Thus  thorium  may  be  changed  from  a  nitrate 
to  a  chloride  or  a  sulphide,  or  it  may  undergo  any  sort  of 
chemical  reaction,  without  any  change  whatever  being  notice- 
able in  its  activity.  Furthermore,  radioactivity  has  been  found 
to  be  independent  of  all  physical  as  well  as  chemical  conditions. 
The  lowest  cold  or  greatest  heat  does  not  appear  to  affect  it 
in  the  least.  Radioactivity,  therefore,  seems  to  be  as  unalter- 
able a  property  of  the  atoms  of  radioactive  substances  as  is 
weight  itself.  For  this  reason  Rutherford  has  advanced  the 
theory  that  the  atoms  of  radioactive  substances  are  slowly 
disintegrating  into  simpler  atoms.  Uranium  and  thorium  have 
the  heaviest  atoms  of  all  the  elements.  For  some  unknown 
reason  they  seem  not  infrequently  to  become  unstable  and 
project  off  a  part  of  their  mass.  This  projected  mass  is  the 
alpha  particle.  What  is  left  of  the  atom  after  the  explosion  is 
a  new  substance  with  chemical  properties  different  from  those 
of  the  original  atom.  This  new  atom  is,  in  general,  also  un- 
stable and  breaks  down  into  something  else.  This  process  is 
repeated  over  and  over  again  until  some  stable  form  of  atom  is 
reached.  Somewhere  in  the  course  of  this  atomic  catastrophe 
some  electrons  leave  the  mass ;  these  are  'beta  rays. 

According  to  this  point  of  view,  which  is  now  generally 
accepted,  radium  is  simply  one  of  the  stages  in  the  disintegra- 
tion of  the  uranium  atom.  The  atomic  weight  of  uranium  is 
238.5,  that  of  radium  about  226,  that  of  helium  3.994.  Radium 
would  then  be  uranium  after  it  has  lost  3  helium  atoms.  The 
further  disintegration  of  radium  through  four  additional  trans- 
formations has  been  traced.  It  has  been  conjectured  that  the 
fifth  and  final  one  is  lead.  If  we  subtract  8  x  3.994  from  238.5, 
we  obtain  206.5,  which  is  very  close  to  the  accepted  value  for 
lead,  namely  207.  In  a  similar  way  six  successive  stages  in 
the  disintegration  of  the  thorium  ?Lom  (atomic  weight  232) 
have  been  found,  but  the  final  product  is  unknown. 


426  INVISIBLE  EADIATIONS 

513.  Energy  stored  up  in  the  atoms  of  the  elements.  In 
1903  the  two  Frenchmen,  Curie  and  Labord,  made  an  epoch- 
making  discovery.  It  was  that  radium  is  continually  evolving 
heat  at  the  rate  of  about  one  hundred  calories  per  hour  per 
gram.  More  recent  measurements  have  given  one  hundred 
eighteen  calories.  This  result  was  to  have  been  anticipated 
from  the  fact  that  the  particles  which  are  continually  flying 
off  from  the  disintegrating  radium  atoms  subject  the  whole 
mass  to  an  incessant  internal  bombardment  which  would  be 
expected  to  raise  its  temperature.  This  measurement  of  the 
exact  amount  of  heat  evolved  per  hour  enables  us  to  estimate 
how  much  heat  energy  is  evolved  in  the  disintegration  of  one 
gram  of  radium.  It  is  about  two  thousand  million  calories  — 
fully  three  hundred  thousand  times  as  much  as  is  evolved 
in  the  combustion  of  .one  gram  of  coal.  Furthermore,  it  is 
not  impossible  that  similar  enormous  quantities  of  energy  are 
locked  up  in  the  atoms  of  all  substances,  existing  there  per- 
haps in  the  form  of  the  kinetic  energy  of  rotation  of  the 
electrons.  The  most  vitally  interesting  question  which  the 
physics  of  the  future  has  to  face  is,  Is  it  possible  for  man  to 
gain  control  of  any  such  store  of  subatomic  energy  and  to  use 
it  for  his  own  ends  ?  Such  a  result  does  not  now  seem  likely 
or  even  possible  ;  and  yet  the  transformations  which  the  study 
of  physics  has  wrought  in  the  world  within  a  hundred  years 
were  once  just  as  incredible  as  this.  In  view  of  what  physics 
has  done,  is  doing,  and  can  yei>  do  for  the  progress  of  the 
world,  can  any  one  be  insensible  either  to  its  value  or  to  its 
fascination  ? 


WILLIAM  CONRAD  RONTGEN, 

MUNICH 
Discoverer  of  X  rays 


ANTOINE  HENRI  BECQUEREL, 
PARIS 

Discoverer  of  radioactivity 


MADAME  CURIE,  UNIVERSITY 

OF  PARIS 
Discoverer  of  radium 


E.  RUTHERFORD,  UNIVERSITY  OF 
MANCHESTER  (ENGLAND) 

Discoverer  of  radioactive  trans- 
formations 


A  GROUP  OF  MODERN  PHYSICISTS 


APPENDIX 


00000 


REVIEW  QUESTIONS  AND  PROBLEMS 

1.  A  cubical  box  20  cm.  on  a  side  is  filled  with  equal  parts  of  mer- 
cury and  water.  What  is  the  entire  force  on  the  inner  surface  of  the  box  ? 

2.  Suppose  a  tube  5  mm.  square  and  200  cm.  long  is  inserted  into 
the  top  of  the  box  mentioned  in  the  previous  problem  and  filled  with 
water,  what  will  the  entire  force  be? 

3.  A  floating  dock  is  shown  in  Fig.  463.    When  the  chambers  c  are 
filled  with  water  the  dock  sinks  until  the  water  line  is  at  A.  The  vessel 
is  then  floated  into  the  dock.  As  soon 

as  it  is  in  place,  the  water  is  pumped 

from  the  chambers  until  the  water 

line  is  as  low  as  B.  Workmen  can  then 

get  at  all  parts  of  the  bottom.  If  each 

of  the  chambers  is  10  'ft.  high  and 

10  ft.  wide,  what  must  be  the  length 

of  the  dock  if  it  is  to  be  available  for 

the    Imperatnr    (Hamburg-American  FIG.  463.   Floating  dock 

Line),  of  50,000  tons'  weight  ? 

4.  The  density  of  stone  is  about  2.5.    If  a  boy  can  lift  120  lb.,  how 
heavy  a  stone  can  he  lift  to  the  surface  of  a  pond  ? 

5.  How  many  cubic  centimeters  of  a  liquid  of  specific  gravity  1.5 
must  be  mixed  with  1 1.  of  a  liquid  of  specific  gravity  .8  to  make  a  mix- 
ture of  specific  gravity  1.3  ? 

6.  A  diver  with  his  diving  suit  weighs  100  kg.    It  requires  15  kg.  of 
lead  to  sink  him.    If  the  density  of  lead  is  11.3,  what  is  the  volume  of 
the  diver  and  his  suit? 

7.  A  body  loses  25  g.  in  water,  23  g.  in  oil,  and  20  g.  in  alcohol.   Find 
the  density  of  the  oil  and  of  the  alcohol. 

8.  A  platinum  ball  weighs  330  g.  in  air,  315  g.  in  water,  and  303  g. 
in  sulphuric  acid.    Find  the  density  of  the  platinum,  the  density  of  the 
acid,  and  the  volume  of  the  ball. 

9.  What  fraction  of  the  total  volume  of  an  irregular  block  of  wood 
of  density  .6  will  float  above  the  surface  of  alcohol  of  density  .8  ? 

10.  What  must  be  the  specific  gravity  of  a  liquid  in  which  a  body 
having  a  specific  gravity  of  6.8  will  float  with  half  its  volume  submerged  ? 

427 


428          REVIEW  QUESTIONS  AND  PROBLEMS 

11.  How  large  a  balloon  filled  with  hydrogen  is  needed  to  raise  a 
weight  of  300  lb.,  including  the  balloon  ?    Explain. 

12.  What  is  Boyle's  law?    A  mass  of  air  3  cc.  in  volume  is  intro- 
duced into  the  space  above  a  barometer  column  which  originally  stands 
at  760  mm.    The  column  sinks  until  it  is  only  570  mm.  high.    Find  the 
volume  now  occupied  by  the  air. 

13.  The  diameters  of  the  piston  and  cylinder  of  a  hydrostatic  press 
are  respectively  3  in.  and  30  in.    The  piston  rod  is  attached  2  ft.  from 
the  fulcrum  of  a  lever  12  ft.  long  (Fig.  12,  p.  17).    What  force  must  be 
applied  at  the  end  of  the  lever  to  make  the  press  exert  a  force  of  5000  lb.  V 

14.  How  high  will  a  lift  pump  raise  water  if  it  is  located  upon  the 
side  of  a  mountain  where  the  barometer  reading  is  71  cm.  ? 

15.  If  the  cylinder  of  an  air  pump  is  ^  the  size  of  the  receiver,  what 
fractional  part  of  the  original  air  will  be  left  after  5  strokes  ? 

16.  A  gas  at  constant  pressure  expands  -^~  of  its  volume  at  0°  C. 
for  every  degree  it  is  raised  above  0°  C.   How  much  will  it  expand  for 
every  degree  F.  above  32°  F.  ? 

17.  A  water  tank  8  ft.  deep,  standing  some  distance  above  the  ground, 
closed  everywhere  except  at  the  top,  is  to  be  emptied.    The  only  means 
of  emptying  it  is  a  flexible  tube. 

(a)  What  is  the  most  convenient  way  of  using  the  tube,  and  how 
could  it  be  set  into  operation  ? 

(b)  How  long  must  the  tube  be  to  empty  the  tank  completely  ? 

18.  If  when  the  barometric  height  is  76  cm.  and  the  temperature  is 
30°  C.  some  water  is  introduced  into  an  air-tight  vessel,  what  will  a 
barometer  in  the  vessel  read? 

19.  A  ball  shot  straight  upward  near  a  pond  was  seen  to  .strike  the 
water  in  10  sec.    How  high  did  it  rise?    What  was  its  initial  speed? 

20.  With  what  velocity  must  a  ball  be  shot  upward  to  rise  to  the 
height  of  the  Washington  Monument  (555  ft.)?    I  low  long  before  it 
will  return? 

21.  A  pull  of  a  dyne  acts  for  3  sec.  on  a  mass  of  1  g.    What  velocity 
does  it  impart? 

22.  How  long  must  a  force  of  100  dynes  act  on  a  mass  of  20  g.  to 
impart  to  it  a  velocity  of  40  cm.  per  second  ? 

23.  A  force  of  1  dyne  acts  on  1  g.  for  1  sec.    How  far  has  the  gram 
been  moved  at  the  end  of  the  second  ? 

24.  A  steamboat  weighing  20,000  metric  tons  is  being  pulled  by  a 
tug  which  exerts  a  pull  of  2  metric  tons.    (A  metric  ton  is  equal  to 
1000  kg.)    If  the  friction  of  the  water  were  negligible,  what  velocity 
would  the  boat  acquire  in  4  min.  ?    (Reduce  mass  to  grains,  force  to 
dynes,  and  remember  that  F  =  mv/t.) 


APPENDIX  429 

25.  If  a  train  of  cars  weighs  200  metric  tons,  and  the  engine  in  pull- 
ing 5  sec.  imparts  to  it  a  velocity  of  2  m.  per  second,  what  is  the  pull  of 
the  engine  in  metric  tons  ? 

26.  A  steel  ball  dropped  into  a  pail  of  moist  clay  from  a  height  of  a 
meter  sinks  to  a  depth  of  2  cm.    How  far  will  it  sink  if  dropped  4m.? 

27.  Neglecting  friction,  find  how  much  force  a  boy  would  have  to 
exert  to  pull  a  100-lb.  waggn  up  an  incline  which  rises  5  ft.  for  every 
100  ft.  of  length  traversed  on  the  incline.    Give  not  merely  the  numeri- 
cal solution  of  the  problem,  but  state  why  you  solve  it  as  you  do,  and 
how  you  know  that  your  solution  is  correct. 

28.  Describe  fully  how  you  would  proceed  to  find  the  density  of  an 
irregular   solid  heavier  than  water,   showing  why  in  every  case  you 
proceed  as  you  do. 

29.  A  rifle  weighing  5  Ib.  discharges  a  4-oz.  bullet  with  a  velocity  of 
100  ft.  per  sec.    What  will  be  the  velocity  of  the  rifle  in  the  opposite 
direction  ? 

30.  A  bullet  weighing  2  oz.  is  shot  into  a  body  weighing  30  Ib.  hang- 
ing freely  suspended.    If  the  velocity  of  the  bullet  is  1500  ft.  per  second, 
what  will  be  the  vertical  height  to  which  the  body  will  be  raised  ? 

31.  How  many  times  as  much  weight  will  a. wire  which  is  twice  as 
thick  as  another  of  similar  material  support? 

32.  A  force  of  3  Ib.  stretches  1  mm.  a  wire  that  is  1  m.  long  and  .1  mm. 
in  diameter.    How  much  force  will  it  take  to  stretch  5  mm.  a  wire  of  the 
same  material  4  in.  long  and  .15  mm.  in  diameter  ? 

33.  Why  do  some  liquids  rise  while  others  are  depressed  in  capillary 
tubes? 

34.  A  metal  rod  230  cm.  long  expanded  2.7 o  mm.  in  being  raised 
from  0°  C.  to  100°  C.    Find  its  coefficient  of  linear  expansion. 

35.  If  iron  rails  are  30  ft.  long,  and  if  the  variation  of  temperature 
throughout  the  year  is  50°  C.,  what  space  must  be  left  between  their  ends  ? 

36.  If  the  total  length  of  the  iron  rods  b,  d,  e,  and  i  in  a  compen- 
sated pendulum  (Fig.  129)  is  2  m.,  what  must  be  the  total  length  of 
the  copper  rods  c  if  the  period  of  the  pendulum  is  independent  of  tem- 
perature ? 

37.  Decide  from  the  table  of  expansion  coefficients  given  on  page  128 
why  the  wires  which  lead  the  current  through  the  walls  of  incandescent 
electric-light  bulbs  are  always  made  of  platinum ;   that  is,  why  it  is 
impossible  to  seal  any  other  metal  into  glass. 

38.  If  a  diver  descends  to  a  depth  of  100  ft.,  what  is  the  pressure  to 
which  he  is  subjected?    What  is  the  density  of  the  air  in  his  suit,  the 
density  at  the   surface  where  the  pressure  is   75  cm.  being   .00123  ? 
(Assume  the  temperature  to  remain  unchanged.) 


430          REVIEW  QUESTIONS  AND  PROBLEMS 

39.  A  bubble  escapes  from  a  diver's  suit  at  a  depth  of  100  ft.  where 
the  temperature  is  4°  C.    To  how  many  times  its  original  volume  has 
the  bubble  grown  by  the  time  it  reaches  the  surface,  where  the  tem- 
perature is  30°  C.  and  the  barometric  height  75  cm.? 

40.  Find  the  density  of  the  air  in  a  furnace  whose  temperature  is 
1000°  C.,  the  density  at  0°  C.  being  .001293. 

41.  The  air  within  a    half-inflated   balloon  occupies   a  volume   of 
100,000  1.    The  temperature  is  15°  C.  and  the  barometric  height  75  cm. 
What  will  be  its  volume  after  the  balloon  has  risen  to  the  height  of 
Mt.  Blanc,  where  the  pressure  is  37  cm.  and  the  temperature  —  10°  C.? 

42.  If  the  volume  of  a  quantity  of  air  at  30°  C.  is  200  cc.,  at  what 
temperature  will  its  volume  be  300  cc.,  the  pressure  remaining  the  same? 

43.  When  the  barometric  height  is  76  cm.  and  the  temperature  0°  C., 
the  density  of  air  is  .001293.    Find  the  density  of  air  when  the  tem- 
perature is  38°  C.  and  the  barometric  height  is  73  cm.  Find  the  density  of 
air  when  the  temperature  is  —  40°  C.  and  the  barometric  height  74  cm. 

44.  A  lever  is  3  ft.  long.    Where  must  the  fulcrum  be  placed  so 
that  a  weight  of  300  Ib.  at  one  end  shall  be  balanced  by  50  Ib.  at 
the  other? 

45.  Where  must  a  load  of  100  Ib.  be  placed  on  a  stick  10  ft.  long,  if 
the  man  who  holds  one  end  is  to  support  30  Ib.,  while  the  man  at  the 
other  end  supports  70  Ib.  ? 

46.  Two  horses  of  unequal  strength  must  be  hitched  as  a  team.    The 
one  is  to  pull  360  Ib.,  while  the  other  pulls  288  Ib.    In  the  doubletree 
50  in.  long,  where  must  the  pin  be  placed  to  permit  an  even  pull  ? 

47.  How  many  gallons  of  water  (8  Ib.  each)  could  a  10  H.P.  engine 
raise  in  one  hour  to  a  height  of  60  ft.  ? 

48.  Find  graphically  the  resultant  of  40  Ib.  N.E.  and  70  Ib.  W. 

49.  In  the  course  of  a  stream  is  a  waterfall  22  ft.  high.    It  is  shown 
by  measurement  that  450  cu.  ft.  of  water  per  second  pour  over  it.    How 
many  foot  pounds  of  energy  could  be  obtained  from  it?   What  horse 
power?  What  becomes  of  this  energy  if  not  used  in  driving  machinery? 

50.  A  car  weighing  60,000  kilos  slides  down  a  grade  which  is  2  m. 
lower  at  the  bottom  than  at  the  top,  and  is  brought  to  rest  at  the 
bottom  by  the  brakes.    How  many  calories  of  heat  are  developed  by 
the  friction? 

51.  A  body  weighing  10  kilos  is  pushed  10  m.  along  a  level  plane.    If 
the  coefficient  of  friction  between  the  block  and  the  plane  is  .125,  how 
many  gram  centimeters  of  work  have  been  done?    How  many  ergs? 
How  many  calories  of  heat  have  been  developed  ? 

52.  Meteorites  are  small  cold  bodies  moving  about  in  space.    Why 
do  they  become  luminous  when  they  enter  the  earth's  atmosphere  ? 


APPENDIX  v      431 

53.  A  piece  of  platinum  weighing  10  g.  is  taken  from  a  furnace  and 
plunged  instantly  into  40  g.  of  water  at  10°  C.    The  temperature  of  the 
water  rises  to  24°  C.   What  was  the  temperature  of  the  furnace  ? 

54.  A  body  is  projected  along  a  horizontal  plane  with  a  velocity  of 
100  ft.  per  second,  the  coefficient  of  friction  being  y^..    How  far  will  it 
go  before  coming  to  rest  ? 

55.  With  what  velocity  must  a  body  be  moving  in  order,  before  com- 
ing to  rest,  to  pass  over  20  m.  on  a  horizontal  plane,  the  coefficient  of 
friction  of  which  is  J  ? 

56.  The  efficiency  of  a  good  condensing  engine  is  about  16%.    How 
much  coal  is  consumed  per  hour  by  a  40,000  H.P.  condensing  engine, 
each  gram  of  coal  being  assumed  to  produce  6000  calories  ? 

57.  The  average  locomotive  has  an  efficiency  of  about  6%.    What 
horse  power  does  it  develop  when  it  is  consuming  1  ton  of  coal  per 
hour?    (See  Problem  56.) 

58.  What  pull  does  a  1000  H.P.  locomotive  exert  when  it  is  running 
at  25  mi.  per  hour  and  exerting  its  full  horse  power  ? 

59.  Equal  weights  of  hot  water  and  ice  are  mixed  and  the  result  is 
water  at  0°  C.    What  was  the  temperature  of  the  hot  water  ? 

60.  From  what  height  must  a  gram  of  ice  at  0°  C.  fall  in  order  to 
melt  itself  by  the  heat  generated  in  the  impact  ? 

61.  What  temperature  will  result  from  mixing  10  g.  of  ice  at  0°  C. 
with  200  g.  of  water  at  25°  C.  ? 

62.  One  hundred  grams  of  water  at  80°  C.  are  thoroughly  mixed  with 
500  g.  of  mercury  at  0°  C.    What  is  the  temperature  of  the  mixture  ? 

63 .  Just  what  will  occur  if  1000  calories  be  applied  to  20  g.  of  ice  at  0°  C.  ? 

64.  If  the  specific  heat  of  lead  is  .031  and  the  mechanical  equivalent 
of  a  calorie  427  g.  m.,  through  how  many  degrees  centigrade  will  a 
1000-g.  lead  ball  be  raised  if  it  falls  from  a  height  of  100  in.,  provided 
all  of  the  heat  developed  by  the  impact  goes  into  the  lead  ? 

65.  How  many  grams  of  ice  must  be  put  into  200  g.  of  water  at  40°  C. 
to  lower  the  temperature  10°  C.? 

66.  Ten  grams  of  steam  at  100°  C.  are  cooled  to  41°  F.    How  much 
heat  is  given  out? 

67.  A  magnetic  pole  of  80  units'  strength  is  20  cm.  distant  from  a 
similar  pole  of  30  units'  strength.    Find  the  force  between  them. 

68.  Two  small  spheres  are  charged  with  +  16  and  —  4  units  of  elec- 
tricity.   With  what  force  will  they  attract  each  other  when  at  a  distance 
of  4  cm.? 

69.  If  the  two  spheres  of  the  previous  problem  are  made  to  touch 
and  are  then  returned  to  their  former  positions,  with  what  force  will 
they  act  on  each  other  ?   Will  this  force  be  attraction  or  repulsion  ? 


432          KEVIEW  QUESTIONS  ANIJ  PROBLEMS 

70.  If  an  electrified  rod  is  brought  near  to  a  pith  ball  suspended  by 
a  silk  thread,  the  ball  is  first  attracted  to  the  rod  and  then  repelled 
from  it.    Explain  this. 

71.  The  diameter  of  No.  20  wire  is  31.90  mils  (1  mil  =  .001  in.)  and 
that  of  No.  30  wire  10.025  mils.     Compare  the  resistances  of   equal 
lengths  of  No.  20  and  No.  30  German-silver  wires. 

72.  What  length  of  No.  30  copper  wire  will  have  the  same  resistance 
as  20  ft.  of  No.  20  copper  wire  ? 

73.  What  length  of  No,  20  German-silver  wire  will  have  the  same 
resistance  as  100  ft.  of  No.  30  copper  wire? 

74.  If  a  certain  Daniell  cell  has  an  internal  resistance  of  2  ohms  and 
an  E.M.F.  of  1.08  volts,  what  current  will  it  send  through  an  ammeter 
whose  resistance  is  negligible  ?    What  current  will  it  send  through  a 
copper  wire  of  2  ohms'  resistance  ?  through  a  German-silver  wire  of 
100  ohms  resistance  ? 

75.  A  Daniell  cell  indicates  a  certain  current  when  connected  to  a 
galvanometer  of  negligible  resistance.    When  a  piece  of  No.  20  German- 
silver  wire  is  inserted  into  the  circuit,  it  is  found  to.  require  a  length  of 
5  ft.  to  reduce  the  current  to  one  half  its  former  value.    Find  the  resist- 
ance of  the  cell  in  ohms,  No.  20  German-silver  wire  having  a  resistance 
of  190.2  ohms  per  1000  ft. 

76.  A  coil  of  unknown  resistance  is  inserted  in  series  with  a  con- 
siderable length  of  No.  30  German-silver  wire  and  joined  to  a  Daniell 
cell.    When  the  terminals  of  a  high-resistance  galvanometer  are  touched 
to  the  wire  at  points  10  ft.  apart,  the  deflection  is  found  to  be  the 
same  as  when  they  are  touched  across  the  terminals  of  the  unknown 
resistance.    What  is  the  resistance  of  the  unknown  coil  ?   (See  §  317, 
p.  252.) 

77.  Find  the  joint  resistance  of  10  ft.  of  No.  30  copper  wire  and  1  ft. 
of  No.  20  German-silver  wire  connected  in  series ;  in  parallel. 

78.  Three  wires,  each  having  a  resistance  of  15  ohms,  were  joined 
abreast  and  a  current  of  3  amperes  sent  through  them.    How  much  was 
the  E.M.F.  of  the  current? 

79.  The  E.M.F.  of  a  certain  battery  is  10  volts  and  the  strength  of 
the  current  obtained  through  an  external  resistance  of  4  ohms  is  1.25 
amperes.    What  is  the  internal  resistance  of  the  battery? 

80.  How  many  cells,  each  of  E.M.F.  1.5  volts  and  internal  resistance 
2  ohms,  will  be  needed  to  send  a  current  of  at  least  1  ampere  through 
an  external  resistance  of  40  ohms? 

81.  How  many  lamps,  each  of  resistance  20  ohms,  and  requiring  a 
current  of  .8  ampere,  can  be  lighted  by  a  dynamo  that  has  an  "output 
of  4000  watts  ? 


APPENDIX  433 

82.  How  many  calories  will  be  developed  in  10  sec.  by  a  current  of 
20  amperes  flowing  through  a  resistance  of  100  ohms  ? 

83.  Draw  a  diagram  of  a  two-station  telegraph  line,  showing  receiving 
and  sending  instruments  at  each  station  and  a  relay  at  one  station. 

84.  Draw  a  diagram  of  an  induction  coil  and  explain  its  action. 

85.  (a)  Describe  and  illustrate  resonance. 

(b)  Find  the  number  of  vibrations  per  second  of  a  fork  which  pro- 
duces resonance  in  a  pipe  1  ft.  long.  (Take  the  speed  of  sound  as  1120 
ft.  per  second.) 

86.  Build  up  a  diatonic  scale  on  C  =  264. 

87.  If  a  vibrating  string  is  found  to  produce  the  note  C  when  stretched 
by  a  force  of  10  lb.,  what  must  be  the  force  exerted  to  cause  it  to  pro- 
duce (a)  the  note  E  ?  (b)  the  note  G  ? 

88.  What  is  meant  by  the  phenomenon  of  beats  in  sound  ?   How  may 
it  be  produced,  and  what  is  its  cause  ? 

89.  Show  what  relation  exists  between  the  wave  length  of  a  note  and 
the  lengths  of  the  shortest  closed  and  open  pipes  which  will  respond  to 
this  note. 

90.  (a)  How  can  you  show  that  the  wave  lengths  of  red  and  green 
lights   are  different,   and  how  can  you   determine  which  one  is  the 
longer  ? 

(b)  Explain  as  well  as  you  can  how  a  telescope  forms  the  image 
which  you  see  when  you  look  into  it. 

91.  A  clapper  strikes  a  bell  once  every  two  seconds.    How  far  from 
the  bell  must  a  man  be  in  order  that  the  clapper  may  appear  to  hit  the 
bell  at  the  exact  instant  at  which  each  stroke  is  heard  ? 

92.  The  note  from  a  piano  string  which  makes  300  vibrations  per 
second  passes  from  indoors,  where  the  temperature  is  20°  C.,  to  outdoors, 
where  it  is  5°C.    What  is  the  difference  in  centimeters  between  the 
wave  lengths  indoors  and  outdoors? 

93.  A  man  riding  on  an  express  train  moving  at  the  rate  of  1  mi. 
per  minute  hears  a  bell  ringing  in  a  tower  in  front  of  him.    If  the  bell 
makes  300  vibrations  per  second,  how  many  pulses  will  strike  his  ear 
per  second,  the  velocity  of  sound  being  1130  ft.  per  second  ?    (The 
number  of  extra  impulses  received  per  second  by  the  ear  is  equal  to  the 
number  of  wave  lengths  contained  in  the  distance  traveled  per  second 
by  the  train.) 

94.  How  many  candles  will  be  required  to  produce  the  same  intensity 
of  illumination  at  2  m.  distance  that  is  produced  by  1  candle  at  30  cm. 
distance  ? 

95.  In  which  medium,  water  or  air,  does  light  travel  the  faster? 
Give  reasons  for  your  answer. 

t 


434          REVIEW  QUESTIONS  AND  PROBLEMS 

96.  Draw  diagrams  to  show  in  what  way  a  beam  of  light  is  bent 
(a)  in  passing  through  a  prism  ;  (b)  in  passing  obliquely  through  a  plate- 
glass  window. 

97.  Distinguish  between  a  real  image  and  a  virtual  image,  and  state 
the  conditions  for  the  formation  of  each  by  a  convex  lens ;  by  a  con- 
cave mirror. 

98.  Show  by  a  diagram  and  explanation  what  is  meant  by  critical 
angle. 

99.  Does  blue  light  travel  slower  or  faster  in  glass  than  red  light? 
How  do  you  know  ? 

100.  Draw  a  figure  to  show  how  a  spectrum  is  formed  by  a  prism, 
and  indicate  the  relative  positions  of  the  red,  the  yellow,  the  green,  and 
the  blue  in  this  spectrum. 

101.  Draw  a  diagram  of  a  slit,  a  prism,  and  a  lens,  so  plac3d  as  to 
form  a  pure  spectrum. 

102.  Why  is  the  order  of  the  colors  in  the  secondary  rainbow  the 
reverse  of  the  order  in  the  primary  bow  ? 

103.  An  object  5  cm.  long  is  50  cm.  from  a  concave  mirror  of  focal 
length  30  cm.    Where  is  the  image,  and  what  is  its  size  ? 

104.  An  object  is  20  cm.  from  a  lens  of  focal  length  30.    Where  is 
the  image  ? 

105.  Why  is  it  necessary  to  use  a  rectifying  crystal  in  series  with  a 
telephone  receiver  to  detect  electric  waves  ? 

106.  Explain  why  an  electroscope  is  discharged  when  a  bit  of  radium 
is  brought  near  it. 


INDEX 


Aberration,  chromatic,  400 

Absolute  temperature,  121 

Absolute  units,  6 

Absorption,  of  gases,  113  ff.;  of  light 
waves,  405 ;  and  radiation,  410 

Acceleration,  denned,  92  ;  of  grav- 
ity, 93 

Achromatic  lens,  400 

Adhesion,  103  ;  effects  of,  109 

Aeronauts,  height  of  ascent  of,  36 

Air,  pressure  of,  27  ;  weight  of,  26  ; 
compressibility  of,  33  ;  expansibil- 
ity of,  34  ;  pump,  40  ;  brake,  46 

Air  brake,  46 

Alternator,  292 

Amalgamation  of  zinc  plate,  257 

Ammeter,  275 

Ampere,  245,  246  ;  definition  of,  269 

Amundsen,  215 

Anode,  267 

Arc  light,  281  ;  automatic  feed  for, 
282 

Archimedes,  principle  of,  21  ;  por- 
trait of,  22 

Armature,  ring  type,  291,  293 ;  drum 
type,  291,  293,  295 

Artesian  wells,  20 

Atmosphere,  pressure  of,  29 ;  extent 
and  character  of,  36;  height  of 
homogeneous,  38  ;  humidity  of,  66 

Atoms,  energy  in,  426 


Back  E.M.F.  in  motors,  298 
Baeyer,  von,  408 


Balance,  7 

Balance  wheel,  129 

Ball  bearings,  154 

Balloon,  44 

Barometer,   mercury,   30;    aneroid, 

31 ;     von    Guericke's,    31 ;     self- 

registering,  36 
Batteries,  primary,  256  ff . ;  storage, 

269 

Beats,  327,  342 
Becquerel,  421,  426 
Bell,  Alexander  Graham,  310 
Bell,  electric,  275 
Bellows,  47 
Bicycle  pedal,  155 
Binocular  vision,  390 
Boiler,  steam,  187 
Boiling  points,    definition   of,    178; 

effect  of  pressure  on,  179 
Boyle's  law,  stated,  35 ;  explained,  53 
Brittleness,  104 
Brooklyn  Bridge,  130 
Brownian  movements,  53 
Bunsen,  369 

Caisson,  45 

Calories,  160 ;  developed  by  electric 
currents,  279 

Camera,  pin-hole,  383;  photographic, 
383 

Candle  power,  defined,  368;  of  in- 
candescent lamps,  281 ;  of  arc 
lamps,  282 

Capacity,  electric,  232 


435 


436 


INDEX 


Capillarity,  105  ff. 

Cartesian  diver,  43 

Cathode,  denned,  267 

Cathode  rays,  417 

Cells,  galvanic,  241  ;  primary,  256  ff. ; 
local  action  in,  257 ;  theory  of, 
258 ;  Daniell,  261 ;  Leclanche",  263 ; 
Weston,  263;  dry,  264;  combina- 
tions of,  265 ;  storage,  269 

Center  of  gravity,  84 

Centrifugal  force,  96 

Charcoal,  absorption  by,  113 

Charles,  law  of,  123 

Chemical  effects  of  currents,  267 

Clouds,  formation  of,  65 

Coefficient  of  expansion,  of  gases, 
123 ;  of  liquids,  126 ;  of  solids,  127  ; 
of  friction,  154 

Coherer,  414 

Cohesion,  103 ;  properties  depending 
on,  104 

Coils,  magnetic  properties  of,  272 ; 
currents  induced  in  rotating,  288 

Cold  storage,  195 

Color,  and  wave  length,  393 ;  of 
bodies,  395 ;  compound,  396 ;  com- 
plementary, 397;  of  thin  films, 
399 ;  of  pigments,  398 

Commutator,  292 

Compass,  216 

Component,  78 

Condensation  of  water  vapor,  64 

Condensers,  235 

Conduction,  of  heat,  197 ;  of  elec- 
tricity, 221 

Conjugate  foci,  372 

Conservation  of  energy,  165 

Convection,  200  ff . 

Cooling,  and  evaporation,  67,  184 ;  of 
a  lake,  127 ;  by  expansion,  163 ; 
artificial,  by  solution,  182 

Cooper-Hewitt  lamp,  283 


Coulomb,  269 

Couple,  138 

Crane,  146 

Critical  angle,  355,  357 

Crookes,  352,  423 

Curie,  426 

Currents,  wind  and  ocean,  201 ; 
measurement  of  electric,  240  ff . ; 
magnetic  fields  about,  244 ;  effects 
of  electric,  267  ff.;  induced  elec- 
tric, 284  ff. 

Curvature,  of  a  liquid  surface,  108  ; 
defined,  364;  of  waves,  371;  of 
mirror,  378  ;  center  of,  382 

Daniell  cell,  261 

Davy  safety  lamp,  199 

Declination,  or  dip,  216 

Density,  defined,  8;  table  of,  8,  9; 
formula  for,  9;  methods  of  find- 
ing, 23  ff . ;  of  air  below  sea  level, 
38 ;  of  saturated  vapor,  62 ;  max- 
imum, of  water,  126;  of  electric 
charge,  228 

Dew,  formation  of  ?  65 ^ 

Dew  point,  66 

Diffusion,  of  gases,  54 ;  of  liquids, 
58 ;  of  solids,  72 ;  of  light,  353 

Discord,  342 

Dispersion,  394 

Dissociation,  258 

Distillation,  180 

Diving  bell,  45 

Diving  suit,  46 

Doppler's  principle,  in  sound,  321; 
in  light,  406 

Ductility,  104 

Dynamo,  principle  of,  284 ;  rule,  287 ; 
alternating-current,  290 ;  four-pole 
direct-current,  294 ;  series-,  shunt-, 
and  compound-wound,  296 

Dyne,  98 


INDEX 


437 


Eccentric,  187 

Echo,  322 

Edison,  310 

Efficiency,  defined,  156;  of  simple 
machines,  156 ;  of  water  motors, 
157,  158;  of  steam  engines,  189; 
of  electric  lights,  281  ff . ;  of  trans- 
formers, 307 

Elasticity,  101 ;  limits  of,  102 

Electric  charge,  unit  of,  221 ;  distri- 
bution of,  227  ;  density  of,  228 

Electrical  machines,  237  ff. 

Electricity,  static,  218  ff.;  two-fluid 
theory  of,  222  ;  current  of,  240  ff . 

Electrolysis  of  water,  267 

Electromagnet,  274 

Electromotive  force,  defined,  247 ; 
of  galvanic  cells,  249;  induced, 
285;  strength  of  induced,  288; 
back,  in  motors,  298  ;  in  secondary 
circuit,  302 ;  at  make  and  break, 
304 

Electron  theory,  223,  418  ff. 

Electrophorus,  236 

Electroplating,  268 

Electroscope,  220,  226 

Electrostatic  voltmeter,  234 

Electrotyping,  269 

Energy,  defined,  148 ;  potential  and 
kinetic,  149;  transformations  of, 
149,  166 ;  formula  for,  151 ;  con- 
servation of,  165 ;  expenditure  of 
electric,  279  ;  stored  in  atoms,  426 

Engine,  steam,  185  ;  compound,  189 ; 
gas,  190 

English  equivalents  of  metric  units, 
5 

Equilibrant,  77 

Equilibrium,  stable,  85  ;  neutral,  87 ; 
unstable,  87 

Erg,  147 

Ether,  358 


Evaporation,  57;  effect  of  temper- 
ature on,  59 ;  effect  of  air  on,  62 ; 
cooling  effect  of,  67 ;  freezing  by, 
68;  effect  of  surface  on,  70;  of 
solids,  71;  and  boiling,  180;  in- 
tense cold  by,  184 

Expansion,  of  gases,  123  ;  of  liquids, 
58,  125 ;  unequal,  of  metals,  129  ; 
cooling  by,  163;  on  solidifying, 
174 

Eye,  385 

Fahrenheit,  118 

Falling  bodies,  88-93 

Faraday,  269,  284 

Fields,  magnetic,  212 

Films,  contractility  of,    106 ;    color 

of,  399 

Fire  syringe,  163 
Float  valve,  140 
Floating  needle,  111 
Focal  length,  of  convex  mirror,  378, 

382  ;  of  convex  lens,  371 
Fog,  formation  of,  65 
Foley,  380 
Force,  definition  of,  74  ;  method  of 

measuring,  75  ;  composition  of,  76  ; 

resultant  of,  76 ;  component  of,  78 ; 

centrifugal,  96 ;  beneath  liquid,  11 ; 

lines  of,  211 ;  fields  of,  212 
Formulas  for  lenses  and  mirrors,  380 
Foucault,  352 
Foucault  currents,  305 
Franklin,  231 
Fraunhof  er  lines,  405 
Freezing  mixtures,  183 
Freezing  points,   table   of,  173 ;    of 

solutions,  183 
Friction,  153  ff. 
Fundamentals,  defined,  335;  in  pipes, 

344 
Fusion,  heat  of,  170 


438 


INDEX 


Galileo,  88, 89, 116, 119 ;  portrait  of,  88 

Galvanic  cell,  241 

Galvanometer,  245,  246 

Gas  engine,  190 

Gas  meter,  47 

Gay-Lussac,  law  of,  123 

Geissler  tubes,  417 

Gilbert,  218 

Gram,  of  mass,  3 ;  of  force,  74 

Gramophone,  349 

Gravitation,  law  of,  83 

Gravity,  variation  of ,  75 ;  center  of,  84 

Guericke,  Otto  von,  31,  32,  40 

Hail,  formation  of,  65 

Hardness,  104 

Harmony,  342 

Hay  scales,  146 

Heat,  mechanical  equivalent  of, 
159  ff. ;  unit  of,  160 ;  produced  by 
friction,  161  ;  produced  by  colli- 
sion, 162;  produced  by  compression, 
162  ;  specific,  167  ;  of  fusion,  170  ; 
latent,  172  ;  transference  of,  197 

Heating,  by  hot  air.  204;  by  hot 
water,  205 

Heating  effects  of  electric  currents, 
279 

Helmholtz,  340 

Henry,  Joseph,  242 

Hertz,  54,  413,  416 

Him,  162 

Hooke's  law,  103 

Horse  power,  147 

Humidity,  66 

Huygens,  358 

Hydraulic  elevator,  18 

Hydraulic  press,  17 

Hydrogen  thermometer,  119 

Hydrometer,  24 

Hydrostatic  paradox,  14 

Hygrometry,  69 


Ice,  manufactured,  194 

Images,  by  convex  lenses,  371  ff.  ; 
in  plane  mirrors,  376  ;  in  convex 
mirrors,  379  ;  size  of,  374  ;  in  con- 
cave mirrors,  379  ;  virtual,  375  ; 
by  concave  lenses,  375 

Incandescent  lighting,  280 

Incidence,  angle  of,  352 

Inclined  plane,  80,  142 

Index  of  refraction,  365 

Induction,  magnetic,  209 ;  electro- 
static, 222;  charging  by,  224;  of 
current,  284 

Induction  coil,  303 

Inertia,  95 

Insect  on  water,  112 

Intensity  of  sound,  321  ;  of  light,  367 

Interference,  of  sound,  328  ;  of  light, 
359 

Ions,  229,  259 

Jackscrew,  143 

Joule,  32,  132,  148,  160  ff.,  307 

Kelvin,  122 

Kilogram,  the  standard,  4 

Kilowatt,  148 

Kinetic  energy,  149,  152 

Kirchhoff,  406 

Laminated  cores,  305 

Lamps,  incandescent,  280  ;  arc,  281 ; 
Cooper-Hewett,  283 

Lantern,  projecting,  384 

Leclanche"  cell,  263 

Lenses,  371  ff.  ;  formula  for,  373 ; 
magnifying  power  of,  386  ;'  achro- 
matic, 400 

Lenz's  law,  285 

Level  of  water,  13 

Lever,  136  ff.  ;  compound,  145 

Leyden  jar,  236 


INDEX 


439 


Light,  speed  of,  351 ;  intensity  of, 
367  ;  diffusion  of,  353  ;  reflection 
of,  352  ;  refraction  of,  398  ;  nature 
of,  358 ;  corpuscular  theory  of, 
358  ;  wave  theory  of,  358  ;  inter- 
ference of,  359  ;  electromagnetic 
theory  of,  416 

Lightning,  231 

Lines,  of  force,  211  ;  isogonic,  216 

Liquid-air  machine,  194 

Liquids,  densities  of,  9  ;  pressure  in, 
13  ;  transmission  of  pressure  by, 
15  ;  incompressibility  of,  33 ;  ex- 
pansion of,  125 

Local  action,  257 

Locomotive,  188 

Loudness  of  sound,  321 

Lyman,  408 

Machine,  liquid-air,  194 

Machines,  general  law  of,  150  ;  effi- 
ciencies of,  156  ;  electrical,  237  ff. 

Magdeburg  hemispheres,  32 

Magnet,  natural,  207  ;  laws  of,  208  ; 
poles  of,  208  ;  electro-,  274 

Magnetism,  207  ff.  ;  nature  of,  213  ; 
theory  of,  214  ;  terrestrial,  215 

Magnifying  power,  of  lens,  386  ;  of 
telescope,  387 ;  of  microscope,  388 ; 
of  opera  glass,  390 

Malleability,  104 

Manometric  flames,  388 

Marconi,  310 

Mass,  unit  of,  4  ;  measurement  of,  6 

Matter,  three  states  of,  72 

Maxwell,  54,  416 

Mechanical  advantage,  135,  137 

Mechanical  equivalent  of  heat,  162 

Melting  points,  table  of,  173  ;  effect 
of  pressure  on,  175 

Meter,  standard,  3 

Michelson,  352 


Microscope,  388 

Mirrors,  376  ff.  ;  convex,  378  ;  con- 
cave, 379  ;  formula  for,  381 

Mixtures,  method  of,  168 

Molecular  constitution  of  matter,  50 

Molecular  forces,  in  solids,  101 ;  in 
liquids,  105 

Molecular  motions,  in  gases,  50 ;  in 
liquids,  57  ;  in  solids,  71 

Molecular  nature  of  magnetism,  213 

Molecular  velocities,  54,  55 

Moments  of,  force,  137 

Momentum  defined,  96 

Morse,  276 

Motion,  uniformly  accelerated,  91; 
laws  of,  92  ;  perpetual,  165 

Motor,  electric,  principle  of,  286 ; 
rule,  287  ;  street-car,  297 

Newton,  law  of  gravitation,  83  ;  por- 
trait of,  96  ;  laws  of  motion  of,  95  ; 
work  principle,  141  ;  and  corpus- 
cular theory,  359 

Niagara,  166,  167 

Nichols,  E.  F.,  408 

Nodes,  in  pipes,  329 ;  in  strings, 
334 

Noise  and  music,  320 

Nonconductors,  of  heat,  199 ;  of 
electricity,  221 

North  magnetic  pole,  215 

Ocean  currents,  201 

Oersted,  242 

Ohm,  252 

Ohm's  law,  252 

Onnes,  Kamerlingh,  69,  122 


Optical  instruments,  383  ff. 
Organ  pipes,  347 
Oscillatory  discharge,  413 
Overtones,  335  ;  in  pipes,  345 


440 


INDEX 


Parachute,  44 

Parallel  connections,  255,  265 

Parallelogram  law,  78 

Pascal,  15,  16,  29 

Pendulum,  force-moving,  81 ;  laws 
of,  94  ;  compensated,  128 

Permeability,  210 

Perpetual  motion,  165 

Perrier,  30 

Phonograph,  349 

Photometers,  367,  369 

Pisa,  tower  of,  88 

Pitch,  cause  of,  320 

Pneumatic  inkstand,  32 

Points,  discharging  effect  of,  228 

Polarization,  of  galvanic  cells,  260  ; 
of  light,  366 

Potential,  defined,  232  ;  unit  of  ,263 ; 
measurement  of,  233,  275 

Power,  definition  of,  147 ;  horse, 
147 

Pressure,  in  liquids,  13  ;  defined,  13  ; 
in  air,  27  ;  amount  of  atmospheric, 
29  ;  of  saturated  vapor,  60  ;  co- 
efficient of  expansion,  120,  123; 
effect  of,  on  freezing,  175  ;  in  pri- 
mary and  secondary,  306 

Pulley,  133  ff. ;  differential,  144 

Pump,  air,  40 ;  water,  41  ;  force,  42 

Quality  of  musical  notes,  337 

Radiation,  thermal,  202  ;  invisible, 
408  ff . ;  and  temperature,  409  ;  and 
absorption,  410  ;  electrical,  412 

Radioactivity,  421 

Radiometer,  408 

Radium,  discovery  of,  421 

Rain,  formation  of,  65 

Rainbow,  401 

Ratchet  wheel,  155 

Rayleighr  352 


Rays,  infra-red,  408 ;  ultra-violet, 
408  ;  cathode,  417  ;  Rontgen,  419  ; 
Becquerel,  422  ;  a,  /3,  and  7,  422  ff . 

Rectifier,  mercury-arc,  309  ;  crystal, 
415 

Refining  of  metal,  269 

Reflection,  of  sound,  322  ;  of  light, 
352  ;  total,  355 

Refraction,  of  light,  354  ;  index  of, 
365  ;  explanation  of,  362 

Regelation,  175 

Relay,  276 

Resistance,  electric,  defined,  251  ; 
specific,  251  ;  table  of,  251  ;  unit 
of,  252  ;  laws  of,  252 ;  internal, 
253 ;  measurement  of,  254 

Resonance,  acoustical,  323  ff . ;  elec- 
trical, 412 

Resonators,  326 

Resultant,  76 

Retentivity,  210 

Retinal  fatigue,  398 

Right-hand  rule,  245,  273 

Rise  of  liquids,  in  exhausted  tubes, 
37  ;  in  capillary  tubes,  108 

Roemer,  351 

Rontgen,  419,  426 

Rowland,  164,  352 

Rubens,  408 

Rumford,  160,  367 

Rutherford,  422,  426 

Saturation,  of  vapors,  54 ;  magnetic, 

215 
Scales,  musical,  331  ;  diatonic,  332  ; 

even-tempered,  333 
Screw,  143 
Self-induction,  302 
Series  connections,  255,  265 
Shadows,  356 
Shunts,  255 
Singing  flame,  342 


INDEX 


441 


Siphon,  explanation  of,  40;  inter- 
mittent, 40 

Snow,  formation  of,  65 

Soap  films,  107,  393 

Solar  spectrum,  404,  406 

Sonometers,  334 

Sound,  sources  of,  314 ;  speed  of, 
315 ;  nature  of,  314 ;  musical,  320 ; 
reflection  of,  322 ;  foci,  323  ;  inter- 
ference of,  328 

Sounder,  276 

Sounding  boards,  326 

Spark,  oscillatory  nature  of,  413  ;  in 
vacuum,  41 7 ;  photography,  413, 41 4 

Spark  length  and  potential,  234 

Speaking  tubes,  321 

Specific  gravity,  9 

Specific  heat,  defined,  167 ;  meas- 
ured, 168  ;  table  of,  169 

Spectra,  401  ff .  ;  continuous,  403 ; 
bright-line,  408  ;  pure,  405 

Spectrum  analysis,  404 

Speed,  of  sound,  315 ;  of  light,  351  ; 
of  light  in  water,  352,  364;  of 
electric  waves,  .413 

Spinthariscope,  424 

Steam  engine,  185  ff. 

Steam  turbine,  192 

Steelyards,  140 

Stereoscope,  390 

Storage  cells,  269 

Strings,  laws  of,  333 

Sublimation,  71 

Sun,  energy  derived  from,  167 

Surface  tension,  106 

Sympathetic  vibrations,  of  sound, 
340 ff.;  electrical,  412 

Telegraph,  276  ff .  ;  wireless  414  ff. 
Telephone,  310  ff. 

Telescope,  astronomical,  387 ;  terres- 
trial, 389 


Temperature,  measurement  of,  116 ; 
absolute,  121  ;  low,  122 

Tenacity,  102 

Thermometer,  Galileo's,  116  ;  mer- 
cury, 117;  Fahrenheit,  118;  gas, 
120,  121  ;  alcohol,  121  ;  maximum 
and  minimum,  122 ;  the  dial, 
130 

Thermoscope,  409 

Thermostat,  129 

Thomson,  419,  420 

Three-color  printing,  399 

Toepler-Holtz,  237 

Torricelli,  experiment  of,  28 

Transformer,  305,  307 

Transmission,  electrical,  308  ;  of 
pressure,  15  ;  of  sound,  316 

Transmitter,  telephone,  311 

Trowbridge,  409 

Turbine,  water,  158  ;  steam,  192 

Units,  of  length,  2  ;  of  area,  2  ;  of 
volume,  2  ;  of  mass,  4  ;  of  time,  6  ; 
three  fundamental,  5;  C.G.S.,  6; 
of  force,  74,  98  ;  of  work,  132  ;  of 
power,  147  ;  of  heat,  160  ;  of  mag- 
netism, 209;  of  potential,  263; 
of  current,  245,  269  ;  of  resistance, 
252  ;  of  light,  368 

Vacuum,  sound   in,  315 ;  spark  in, 

417 

Vaporization,  heat  of,  177,  178 
Velocity,   of   falling  body,    91  ;    of 

sound,  315  ;  of  light,  351 
Ventilation,  203 
Vibration,  numbers,  331  ;  of  strings, 

333  ;    forced,    326  ;    sympathetic, 

340  ff. 
Vision,  distance    of    most   distinct, 

386 
Visual  angle,  385 


442 


INDEX 


Volt,  234,  263,  288 
Volta,  234 

Voltmeter,  249,  275;  electrostatic, 
234 

Watch,  balance  wheel  of,  129 ;  wind- 
ing mechanism  of,  147 

Water,  density  of,  4  ;  city  supply  of, 
19 ;  maximum  density  of,  126 ; 
expansion  of,  on  freezing,  200 

Water  wheels,  157 

Watt,  148 

Watt,  James,  147,  148 

Wave  length,  defined,  317  ;  formula 
for,  318  ;  of  yellow  light,  362  ;  of 
other  lights,  393 

Wave  theory  of  light,  358 

Waves,  condensational,  318  ;  water, 
319 ;  longitudinal  and  transverse, 
320  ;  light,  transverse,  365 ;  elec- 
tric, 413 

Weighing,  method  of  substitution,  7 


Wet-and-dry-bulb  hygrometer,  69 
Wheel,   and   axle,   142 ;   gear,  144 ; 

worm,  144 ;  water,  157 
White  light,  nature  of,  394 
Wilson,  C.  T.  R.,  424 
Wimshurst  electrical  machine,  238 
Wind  instruments,  344 
Windlass,  145,  146 
Winds,  201 

Wireless  telegraphy,  414 
Work,  defined,  131  ;  units  of,  132 ; 

principle  of,  141 
Wright,  Orville,  310 

X  rays,  419 

Yale  lock,  141 

Yard,  2 

Yerkes  telescope,  388 

Zeiss  binocular,  391 
Zeppelin  airship,  44 


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